WEBVTT 00:00:00.000 --> 00:00:01.920 align:middle line:90% [SQUEAKING] 00:00:01.920 --> 00:00:04.320 align:middle line:90% [RUSTLING] 00:00:04.320 --> 00:00:05.760 align:middle line:90% [CLICKING] 00:00:05.760 --> 00:00:11.820 align:middle line:90% 00:00:11.820 --> 00:00:13.640 align:middle line:84% PROFESSOR: So last time, just quickly 00:00:13.640 --> 00:00:19.040 align:middle line:84% recall, we looked at a function on some interval 00:00:19.040 --> 00:00:25.740 align:middle line:84% into R. We looked at a partition, P. So here, 00:00:25.740 --> 00:00:28.360 align:middle line:84% we have the interval from a to b. 00:00:28.360 --> 00:00:35.760 align:middle line:84% Then we have a partition, x0, x1, up to, say, xn. 00:00:35.760 --> 00:00:41.820 align:middle line:84% We looked at Mi, which was the sup, on this little interval 00:00:41.820 --> 00:00:45.960 align:middle line:90% here from xi minus 1 to xi of f. 00:00:45.960 --> 00:00:51.800 align:middle line:84% We looked at little mi, which was the inf. 00:00:51.800 --> 00:00:53.150 align:middle line:90% f here was bounded again. 00:00:53.150 --> 00:00:57.920 align:middle line:90% 00:00:57.920 --> 00:01:03.820 align:middle line:84% And inf on the same interval, xi minus 1 to xi. 00:01:03.820 --> 00:01:07.460 align:middle line:84% And then we looked at the upper and the lower sum. 00:01:07.460 --> 00:01:11.580 align:middle line:84% So we looked at the upper and lower sum 00:01:11.580 --> 00:01:13.940 align:middle line:90% with respect to this partition. 00:01:13.940 --> 00:01:19.700 align:middle line:84% So this here was the sum of Mi and then delta xi 00:01:19.700 --> 00:01:24.300 align:middle line:84% and the lower sum for this partition, which 00:01:24.300 --> 00:01:31.620 align:middle line:84% was the sum of this smaller mi's delta xi, where the delta xi was 00:01:31.620 --> 00:01:37.860 align:middle line:84% the length of these little intervals here, like that. 00:01:37.860 --> 00:01:40.620 align:middle line:84% And then it was Riemann integrable. 00:01:40.620 --> 00:01:48.140 align:middle line:84% Then we looked at the lower and the upper Riemann integral. 00:01:48.140 --> 00:01:54.580 align:middle line:84% So we looked at the upper and lower Riemann integral. 00:01:54.580 --> 00:02:01.920 align:middle line:84% This here was now the inf of these upper integrals 00:02:01.920 --> 00:02:03.760 align:middle line:90% over all partitions. 00:02:03.760 --> 00:02:12.840 align:middle line:84% And the lower one, that was the sup of the lower 00:02:12.840 --> 00:02:17.080 align:middle line:90% sums over all partitions. 00:02:17.080 --> 00:02:27.360 align:middle line:84% And then we said that f is Riemann integrable if those two 00:02:27.360 --> 00:02:30.460 align:middle line:84% numbers are the same, if the upper one here, 00:02:30.460 --> 00:02:33.200 align:middle line:84% which is always bigger or equal to the lower one, 00:02:33.200 --> 00:02:37.960 align:middle line:84% if that is actually equal to the lower one, the lower Riemann 00:02:37.960 --> 00:02:40.328 align:middle line:90% integral. 00:02:40.328 --> 00:02:43.300 align:middle line:90% And if this is the case-- 00:02:43.300 --> 00:02:45.880 align:middle line:90% so this here is-- 00:02:45.880 --> 00:02:48.000 align:middle line:84% it's Riemann integrable if and only 00:02:48.000 --> 00:02:49.920 align:middle line:84% if these two numbers are the same. 00:02:49.920 --> 00:02:53.120 align:middle line:84% And in that case, we wrote-- for this common number, 00:02:53.120 --> 00:02:55.040 align:middle line:90% we wrote just the-- 00:02:55.040 --> 00:02:59.020 align:middle line:84% we wrote it as the Riemann integral. 00:02:59.020 --> 00:03:06.000 align:middle line:84% So now, there was this parallel to things we have done before. 00:03:06.000 --> 00:03:09.860 align:middle line:84% There was these algebraic rules that we talked about 00:03:09.860 --> 00:03:11.040 align:middle line:90% at the very end. 00:03:11.040 --> 00:03:12.705 align:middle line:90% So what were those? 00:03:12.705 --> 00:03:15.740 align:middle line:90% 00:03:15.740 --> 00:03:24.060 align:middle line:84% If you take a function that is Riemann integrable, 00:03:24.060 --> 00:03:31.140 align:middle line:84% and you take c is a constant, then the product here 00:03:31.140 --> 00:03:32.920 align:middle line:90% is also Riemann integrable. 00:03:32.920 --> 00:03:37.180 align:middle line:90% 00:03:37.180 --> 00:03:43.020 align:middle line:84% And the integral of the product is just-- 00:03:43.020 --> 00:03:46.500 align:middle line:84% product function is just the product 00:03:46.500 --> 00:03:53.020 align:middle line:84% of c with the Riemann integral of f. 00:03:53.020 --> 00:03:55.360 align:middle line:90% So that was 1. 00:03:55.360 --> 00:04:00.480 align:middle line:84% 2 is that if you have a second function, so in addition to f, 00:04:00.480 --> 00:04:07.746 align:middle line:84% you also have a g, and they're both Riemann integrable, 00:04:07.746 --> 00:04:12.220 align:middle line:84% then the sum here is also Riemann integrable. 00:04:12.220 --> 00:04:15.840 align:middle line:90% 00:04:15.840 --> 00:04:25.040 align:middle line:84% And the integral of the sum is equal to-- 00:04:25.040 --> 00:04:26.480 align:middle line:90% from a to b-- 00:04:26.480 --> 00:04:28.930 align:middle line:84% is equal to the sum of the Riemann integral. 00:04:28.930 --> 00:04:31.960 align:middle line:90% 00:04:31.960 --> 00:04:33.420 align:middle line:90% Let's prove this one here. 00:04:33.420 --> 00:04:36.200 align:middle line:90% 00:04:36.200 --> 00:04:38.240 align:middle line:90% I won't prove the first one. 00:04:38.240 --> 00:04:40.400 align:middle line:90% But let's prove the second one. 00:04:40.400 --> 00:04:42.600 align:middle line:90% So this is 2. 00:04:42.600 --> 00:04:45.160 align:middle line:84% Now, to prove this, let me remind you 00:04:45.160 --> 00:04:53.860 align:middle line:84% about one of the lemmas that we had last time. 00:04:53.860 --> 00:04:58.060 align:middle line:84% We were talking about that this was a useful lemma 00:04:58.060 --> 00:05:01.900 align:middle line:84% to determine whether a function was Riemann integrable. 00:05:01.900 --> 00:05:07.860 align:middle line:84% So recall this from last time, that you 00:05:07.860 --> 00:05:14.280 align:middle line:84% have the following lemma, that f, again, is bounded. 00:05:14.280 --> 00:05:21.940 align:middle line:84% So f is Riemann integrable if and only 00:05:21.940 --> 00:05:28.260 align:middle line:84% if, for all epsilon greater than 0, 00:05:28.260 --> 00:05:32.180 align:middle line:84% there exists a partition of this interval here 00:05:32.180 --> 00:05:36.660 align:middle line:84% such that the upper sum, which is always 00:05:36.660 --> 00:05:46.260 align:middle line:84% bigger than the lower sum, this one here, minus the lower sum-- 00:05:46.260 --> 00:05:49.100 align:middle line:84% so this number is always positive. 00:05:49.100 --> 00:05:55.120 align:middle line:84% But if, for all epsilon, you can make this difference here 00:05:55.120 --> 00:05:58.600 align:middle line:84% smaller, then you can find a partition so that you have that. 00:05:58.600 --> 00:06:00.060 align:middle line:90% You proved that last time. 00:06:00.060 --> 00:06:02.600 align:middle line:90% 00:06:02.600 --> 00:06:05.540 align:middle line:84% So now, let's go to the proof of this. 00:06:05.540 --> 00:06:09.920 align:middle line:90% 00:06:09.920 --> 00:06:11.180 align:middle line:90% So proof of 2. 00:06:11.180 --> 00:06:18.520 align:middle line:90% 00:06:18.520 --> 00:06:28.280 align:middle line:84% So given epsilon greater than 0, since f here 00:06:28.280 --> 00:06:34.160 align:middle line:84% is Riemann integrable, then there 00:06:34.160 --> 00:06:45.920 align:middle line:84% exists a partition, P1, such that the upper sum 00:06:45.920 --> 00:06:54.320 align:middle line:84% minus the lower sum is less than epsilon over 2. 00:06:54.320 --> 00:06:58.140 align:middle line:84% This is because the function f was Riemann integrable. 00:06:58.140 --> 00:07:07.620 align:middle line:84% And likewise-- likewise, since g is Riemann integrable, 00:07:07.620 --> 00:07:19.220 align:middle line:84% so there exists a second partition, P2, so that, for g, 00:07:19.220 --> 00:07:21.980 align:middle line:90% you have the same thing-- 00:07:21.980 --> 00:07:23.160 align:middle line:90% for g and P2. 00:07:23.160 --> 00:07:25.300 align:middle line:90% Sorry, this is g. 00:07:25.300 --> 00:07:29.020 align:middle line:84% This upper sum minus the lower sum 00:07:29.020 --> 00:07:33.460 align:middle line:84% for g with respect to this second partition, 00:07:33.460 --> 00:07:35.700 align:middle line:84% this here, is less than epsilon over 2. 00:07:35.700 --> 00:07:40.140 align:middle line:90% 00:07:40.140 --> 00:07:42.130 align:middle line:90% So now you let P here-- 00:07:42.130 --> 00:07:46.080 align:middle line:90% 00:07:46.080 --> 00:07:58.600 align:middle line:84% so now let P be the partition that 00:07:58.600 --> 00:08:12.560 align:middle line:84% has all the dividing points of P1, 00:08:12.560 --> 00:08:30.640 align:middle line:84% together with all the dividing points of P2. 00:08:30.640 --> 00:08:38.509 align:middle line:84% So it has the partitions that are in either one of the two, 00:08:38.509 --> 00:08:40.500 align:middle line:90% or both. 00:08:40.500 --> 00:08:44.580 align:middle line:84% And so now we know that-- so this just means that-- 00:08:44.580 --> 00:08:57.740 align:middle line:84% so P here is a refinement of both P1 and P2. 00:08:57.740 --> 00:09:00.380 align:middle line:90% 00:09:00.380 --> 00:09:02.260 align:middle line:84% So this means that actually, when 00:09:02.260 --> 00:09:04.140 align:middle line:84% you're looking at this, if you're 00:09:04.140 --> 00:09:10.800 align:middle line:84% looking at this upper sum minus this lower sum, 00:09:10.800 --> 00:09:13.860 align:middle line:84% remember that if you go to a refinement, 00:09:13.860 --> 00:09:16.020 align:middle line:90% then this here goes down. 00:09:16.020 --> 00:09:17.440 align:middle line:90% And this here goes up. 00:09:17.440 --> 00:09:19.460 align:middle line:84% So it means that this difference here 00:09:19.460 --> 00:09:25.190 align:middle line:84% would be less than or equal to what it is for P1. 00:09:25.190 --> 00:09:32.980 align:middle line:90% 00:09:32.980 --> 00:09:35.600 align:middle line:84% And this was, of course, less than epsilon over 2. 00:09:35.600 --> 00:09:37.060 align:middle line:90% That was how it was chosen. 00:09:37.060 --> 00:09:41.660 align:middle line:84% And similarly, if you're looking at the upper sum 00:09:41.660 --> 00:09:52.800 align:middle line:84% for g with respect to P, P here is a refinement of P2 also. 00:09:52.800 --> 00:09:56.800 align:middle line:84% So this means that you have that this difference here 00:09:56.800 --> 00:10:04.160 align:middle line:84% is less or equal to the upper sum of g with respect 00:10:04.160 --> 00:10:09.880 align:middle line:84% to P2 minus this lower sum with respect to P2. 00:10:09.880 --> 00:10:11.900 align:middle line:84% And that's less than epsilon over 2. 00:10:11.900 --> 00:10:14.800 align:middle line:90% 00:10:14.800 --> 00:10:19.760 align:middle line:84% But this means now that if you now take the-- 00:10:19.760 --> 00:10:21.110 align:middle line:90% so now, if you take-- 00:10:21.110 --> 00:10:29.600 align:middle line:90% 00:10:29.600 --> 00:10:36.000 align:middle line:84% so now, if you take an interval here of-- 00:10:36.000 --> 00:10:41.300 align:middle line:90% this is from xi minus 1 to xi. 00:10:41.300 --> 00:10:45.080 align:middle line:84% If you're looking at this sum here-- 00:10:45.080 --> 00:10:46.980 align:middle line:84% you have to be a little bit careful here. 00:10:46.980 --> 00:10:50.700 align:middle line:84% But if you're looking at the sum on this integral, 00:10:50.700 --> 00:10:56.140 align:middle line:84% the sum here is less or equal to the sup of what f is-- 00:10:56.140 --> 00:11:01.580 align:middle line:90% this is on this interval here. 00:11:01.580 --> 00:11:04.660 align:middle line:84% On this interval, this function here 00:11:04.660 --> 00:11:12.195 align:middle line:84% is less than the sup of f plus the sup of g 00:11:12.195 --> 00:11:13.070 align:middle line:90% on the same interval. 00:11:13.070 --> 00:11:20.276 align:middle line:90% 00:11:20.276 --> 00:11:25.380 align:middle line:84% And it's, of course, bigger or equal to what-- 00:11:25.380 --> 00:11:38.810 align:middle line:84% it is the inf of f plus the inf of g on that same interval. 00:11:38.810 --> 00:11:42.960 align:middle line:90% 00:11:42.960 --> 00:11:45.020 align:middle line:84% Why do I say you have to be a little bit careful 00:11:45.020 --> 00:11:53.480 align:middle line:84% is that the inf of the sum could be strictly larger than this. 00:11:53.480 --> 00:11:55.840 align:middle line:84% And it could be strictly less than that. 00:11:55.840 --> 00:11:58.720 align:middle line:84% But you certainly have this statement. 00:11:58.720 --> 00:12:04.180 align:middle line:84% Just thinking about, if f and g was the same up to a sign, 00:12:04.180 --> 00:12:09.920 align:middle line:84% if g was like minus f, then the sum would be dead 0. 00:12:09.920 --> 00:12:13.800 align:middle line:84% But one of the functions may be negative on this interval, 00:12:13.800 --> 00:12:15.300 align:middle line:90% and the other one was positive. 00:12:15.300 --> 00:12:18.720 align:middle line:84% So you see, for the sum here, could be dead 0. 00:12:18.720 --> 00:12:21.140 align:middle line:84% But these two numbers, one would be positive. 00:12:21.140 --> 00:12:22.600 align:middle line:90% The other would be negative. 00:12:22.600 --> 00:12:24.960 align:middle line:84% But you certainly have this inequality. 00:12:24.960 --> 00:12:30.240 align:middle line:84% And so, of course, this implies that on this interval, 00:12:30.240 --> 00:12:39.380 align:middle line:84% the sup of the sum here is less than the sup 00:12:39.380 --> 00:12:42.590 align:middle line:90% of the sum of the sups-- 00:12:42.590 --> 00:12:49.100 align:middle line:90% 00:12:49.100 --> 00:12:51.280 align:middle line:84% sum of the sups on the same intervals-- 00:12:51.280 --> 00:12:58.060 align:middle line:90% 00:12:58.060 --> 00:13:01.660 align:middle line:90% and likewise for the inf. 00:13:01.660 --> 00:13:05.900 align:middle line:90% 00:13:05.900 --> 00:13:07.680 align:middle line:90% So you have this thing here. 00:13:07.680 --> 00:13:09.780 align:middle line:84% And you have that this thing here 00:13:09.780 --> 00:13:14.900 align:middle line:84% is clearly bigger or equal to the inf of the same function, 00:13:14.900 --> 00:13:20.740 align:middle line:84% which is the sum of the two, on this interval here. 00:13:20.740 --> 00:13:25.680 align:middle line:84% And that is, of course, bigger than this sum here. 00:13:25.680 --> 00:13:33.380 align:middle line:90% 00:13:33.380 --> 00:13:36.560 align:middle line:90% So this here, this number here-- 00:13:36.560 --> 00:13:42.640 align:middle line:84% you see, this here is like Mi for the function f. 00:13:42.640 --> 00:13:45.760 align:middle line:90% This is what this number is. 00:13:45.760 --> 00:13:48.520 align:middle line:84% And this here is like-- this number here 00:13:48.520 --> 00:13:54.760 align:middle line:84% is like capital Mi for the function g. 00:13:54.760 --> 00:13:58.480 align:middle line:84% And this here-- this here is like little-- 00:13:58.480 --> 00:14:02.600 align:middle line:84% this here is like little mi for the function f. 00:14:02.600 --> 00:14:08.080 align:middle line:84% And this number here is little mi for the function g. 00:14:08.080 --> 00:14:15.880 align:middle line:84% So what we have here is that mi for f plus-- 00:14:15.880 --> 00:14:18.820 align:middle line:90% sorry, let me write this-- 00:14:18.820 --> 00:14:22.720 align:middle line:90% 00:14:22.720 --> 00:14:24.020 align:middle line:90% let me write it over here. 00:14:24.020 --> 00:14:35.140 align:middle line:90% 00:14:35.140 --> 00:14:39.380 align:middle line:84% So what we have is that we have little mi for the function 00:14:39.380 --> 00:14:44.500 align:middle line:84% f plus little mi for the function g 00:14:44.500 --> 00:14:50.980 align:middle line:84% is less than or equal to little mi for the sum, which 00:14:50.980 --> 00:14:56.340 align:middle line:84% is, of course, less than the sup of the same sum 00:14:56.340 --> 00:14:57.520 align:middle line:90% on the same interval. 00:14:57.520 --> 00:15:00.660 align:middle line:90% So this is this. 00:15:00.660 --> 00:15:06.540 align:middle line:84% And that's less or equal to the sup of f on that 00:15:06.540 --> 00:15:12.120 align:middle line:90% interval plus this here of g. 00:15:12.120 --> 00:15:15.260 align:middle line:90% So you have that inequality. 00:15:15.260 --> 00:15:18.080 align:middle line:84% But now you see that this means, of course, 00:15:18.080 --> 00:15:19.900 align:middle line:90% that the contribution-- 00:15:19.900 --> 00:15:26.796 align:middle line:84% so if you're looking at this upper sum for this-- 00:15:26.796 --> 00:15:31.640 align:middle line:84% this upper sum for the sum of these two functions with respect 00:15:31.640 --> 00:15:37.960 align:middle line:84% to P, this here, the contribution you get 00:15:37.960 --> 00:15:40.720 align:middle line:90% is, for the upper, you get this. 00:15:40.720 --> 00:15:42.850 align:middle line:84% And for the lower, you would get-- 00:15:42.850 --> 00:15:45.360 align:middle line:90% 00:15:45.360 --> 00:15:50.480 align:middle line:84% you'd get this here times the length of the interval, right? 00:15:50.480 --> 00:15:57.520 align:middle line:84% So this is-- so again, so what you get on that little interval, 00:15:57.520 --> 00:16:04.880 align:middle line:84% the contribution you get is like Mi f plus g times the length 00:16:04.880 --> 00:16:06.270 align:middle line:90% of the interval-- 00:16:06.270 --> 00:16:13.240 align:middle line:84% this is this-- minus Mi f plus g times the length 00:16:13.240 --> 00:16:14.400 align:middle line:90% of the interval. 00:16:14.400 --> 00:16:18.320 align:middle line:84% This is the contribution that you get here 00:16:18.320 --> 00:16:23.200 align:middle line:84% from this interval-- from this interval, right? 00:16:23.200 --> 00:16:27.660 align:middle line:84% And because this number here is less than that, 00:16:27.660 --> 00:16:42.100 align:middle line:84% then this here is less than Mi f plus Mi g times delta xi minus-- 00:16:42.100 --> 00:16:45.740 align:middle line:84% and then this thing here, you see, 00:16:45.740 --> 00:16:48.620 align:middle line:84% this spreads between these two here. 00:16:48.620 --> 00:16:50.740 align:middle line:84% Bounds are spread between those two. 00:16:50.740 --> 00:17:00.300 align:middle line:84% So this is less than mi f plus little mi g times delta xi. 00:17:00.300 --> 00:17:04.420 align:middle line:90% 00:17:04.420 --> 00:17:06.560 align:middle line:84% And now you're summing over everything. 00:17:06.560 --> 00:17:09.800 align:middle line:84% So this means that this thing here is-- 00:17:09.800 --> 00:17:13.880 align:middle line:84% so this-- and if you just split this here, this sum here, 00:17:13.880 --> 00:17:16.500 align:middle line:84% you can just split that into two parts, 00:17:16.500 --> 00:17:24.279 align:middle line:84% the Mi f times delta xi minus little mi 00:17:24.279 --> 00:17:31.810 align:middle line:84% f times delta xi, and then plus capital Mi 00:17:31.810 --> 00:17:42.200 align:middle line:84% g delta xi minus little mi g delta xi. 00:17:42.200 --> 00:17:47.120 align:middle line:84% So when you're summing these up, you 00:17:47.120 --> 00:17:54.040 align:middle line:84% get a U f, P. That's from this here. 00:17:54.040 --> 00:17:59.216 align:middle line:84% When you're summing these up, U f, P minus, sorry, L f, 00:17:59.216 --> 00:18:04.000 align:middle line:84% P. When you're summing this up, then you're 00:18:04.000 --> 00:18:18.280 align:middle line:90% getting U g, P minus L g, P. 00:18:18.280 --> 00:18:20.470 align:middle line:90% And so what you get here is-- 00:18:20.470 --> 00:18:23.240 align:middle line:90% 00:18:23.240 --> 00:18:25.740 align:middle line:84% again, when you're summing it up, 00:18:25.740 --> 00:18:31.460 align:middle line:84% this here gives you-- when you sum these up, then you get this. 00:18:31.460 --> 00:18:34.600 align:middle line:84% When you sum this here up, you get that. 00:18:34.600 --> 00:18:36.120 align:middle line:84% When you sum this up, you get that. 00:18:36.120 --> 00:18:38.060 align:middle line:84% So summing everything up, you get 00:18:38.060 --> 00:18:43.980 align:middle line:84% U f plus g, comma, this partition, 00:18:43.980 --> 00:18:51.380 align:middle line:84% minus L f plus g with respect to this partition, 00:18:51.380 --> 00:18:55.980 align:middle line:84% is less or equal to U f with respect 00:18:55.980 --> 00:19:01.540 align:middle line:84% to this partition, minus L f with respect to partition P, 00:19:01.540 --> 00:19:13.580 align:middle line:84% plus U g, P, minus L g, P. So you're getting that. 00:19:13.580 --> 00:19:19.540 align:middle line:84% And this thing here, it was a refinement of the partition P1. 00:19:19.540 --> 00:19:23.140 align:middle line:84% So this spread here is less than what it was for P1. 00:19:23.140 --> 00:19:26.800 align:middle line:84% So this here is less than epsilon over 2. 00:19:26.800 --> 00:19:31.160 align:middle line:84% And likewise, this here is a refinement of the partition P2. 00:19:31.160 --> 00:19:34.640 align:middle line:84% So the spread here is less than what it is for P2. 00:19:34.640 --> 00:19:36.600 align:middle line:90% So you're getting this. 00:19:36.600 --> 00:19:38.440 align:middle line:90% And so this here is epsilon. 00:19:38.440 --> 00:19:41.040 align:middle line:90% So this gives you the partition. 00:19:41.040 --> 00:19:43.520 align:middle line:84% And so there's two things you have 00:19:43.520 --> 00:19:49.617 align:middle line:84% to be careful with, that for one function, you get one partition. 00:19:49.617 --> 00:19:51.700 align:middle line:84% For the other function, you get another partition. 00:19:51.700 --> 00:19:55.160 align:middle line:84% So you're looking at the subpartition 00:19:55.160 --> 00:19:59.040 align:middle line:84% that has all of the dividing points of both of them. 00:19:59.040 --> 00:20:02.420 align:middle line:84% And the other things you have to be careful is about-- 00:20:02.420 --> 00:20:07.960 align:middle line:84% and that was embedded here, but it was actually more clearly 00:20:07.960 --> 00:20:10.160 align:middle line:84% on one of the other boards, is that when 00:20:10.160 --> 00:20:13.160 align:middle line:84% you're looking at two functions, and you're 00:20:13.160 --> 00:20:15.200 align:middle line:84% looking at a little interval, and you're 00:20:15.200 --> 00:20:19.160 align:middle line:84% taking the sup of the sum, then that's 00:20:19.160 --> 00:20:21.460 align:middle line:90% bounded by the sum of the sup. 00:20:21.460 --> 00:20:25.600 align:middle line:84% But it's not equal to the sum, and likewise for the inf. 00:20:25.600 --> 00:20:26.100 align:middle line:90% Yeah? 00:20:26.100 --> 00:20:30.260 align:middle line:84% AUDIENCE: [INAUDIBLE] for a partition because you always 00:20:30.260 --> 00:20:32.940 align:middle line:90% have to be [INAUDIBLE]? 00:20:32.940 --> 00:20:36.580 align:middle line:84% PROFESSOR: I mean, so there's not. 00:20:36.580 --> 00:20:39.580 align:middle line:84% But the reason it's not is that once you 00:20:39.580 --> 00:20:43.580 align:middle line:84% have this general framework, you wouldn't really get back 00:20:43.580 --> 00:20:45.620 align:middle line:90% to the partition, actually. 00:20:45.620 --> 00:20:49.640 align:middle line:90% So if it was-- 00:20:49.640 --> 00:20:52.700 align:middle line:84% I mean, once you have this solid , 00:20:52.700 --> 00:20:58.360 align:middle line:90% foundations then you just write. 00:20:58.360 --> 00:20:59.860 align:middle line:84% You just write the integrals, right? 00:20:59.860 --> 00:21:01.100 align:middle line:90% Yeah. 00:21:01.100 --> 00:21:10.800 align:middle line:84% And so, just on that note, this dx, this is-- 00:21:10.800 --> 00:21:24.480 align:middle line:90% so this here-- this here is-- 00:21:24.480 --> 00:21:26.480 align:middle line:90% this here is like-- 00:21:26.480 --> 00:21:28.860 align:middle line:84% it's referring-- this is like a giant S. 00:21:28.860 --> 00:21:32.440 align:middle line:90% So it's referring to summation. 00:21:32.440 --> 00:21:36.200 align:middle line:90% This here is, of course, a d. 00:21:36.200 --> 00:21:38.660 align:middle line:84% And delta is the Greek letter for d. 00:21:38.660 --> 00:21:41.280 align:middle line:84% And so this here, you should think about this here 00:21:41.280 --> 00:21:47.500 align:middle line:84% as referring to some sort of limit of this delta xi. 00:21:47.500 --> 00:21:51.200 align:middle line:90% 00:21:51.200 --> 00:22:01.240 align:middle line:84% So that was-- let me just talk a little bit about the other two 00:22:01.240 --> 00:22:02.320 align:middle line:90% rules. 00:22:02.320 --> 00:22:04.570 align:middle line:90% So one of the other ones is-- 00:22:04.570 --> 00:22:07.880 align:middle line:90% 00:22:07.880 --> 00:22:12.150 align:middle line:84% one of the other ones is that if you take two functions, f and g 00:22:12.150 --> 00:22:12.650 align:middle line:90% again-- 00:22:12.650 --> 00:22:15.200 align:middle line:90% 00:22:15.200 --> 00:22:19.860 align:middle line:84% so number 3 here was that if you have f and g, 00:22:19.860 --> 00:22:24.540 align:middle line:84% they are both integrable on this interval here. 00:22:24.540 --> 00:22:28.380 align:middle line:84% And f here is less or equal to g. 00:22:28.380 --> 00:22:32.120 align:middle line:84% Then it implies that the integral here is less 00:22:32.120 --> 00:22:38.240 align:middle line:90% or equal to the integral of g. 00:22:38.240 --> 00:22:41.220 align:middle line:90% 00:22:41.220 --> 00:22:42.940 align:middle line:90% I won't prove this. 00:22:42.940 --> 00:22:45.400 align:middle line:90% It's sort of, in a way-- 00:22:45.400 --> 00:22:47.580 align:middle line:84% maybe I'll just say words, that if you 00:22:47.580 --> 00:22:51.440 align:middle line:84% take a partition, then, of course, 00:22:51.440 --> 00:22:54.300 align:middle line:84% you're looking at some xi minus 1 00:22:54.300 --> 00:22:57.780 align:middle line:90% and xi, some interval like this. 00:22:57.780 --> 00:23:01.020 align:middle line:84% And so, if you're looking at the function f, 00:23:01.020 --> 00:23:03.300 align:middle line:84% and you're looking at the sup of the function 00:23:03.300 --> 00:23:10.380 align:middle line:84% f on this little interval, well then, obviously, this sup here 00:23:10.380 --> 00:23:15.220 align:middle line:84% is less than the sup of g, just because you have this inequality 00:23:15.220 --> 00:23:17.240 align:middle line:90% everywhere. 00:23:17.240 --> 00:23:19.480 align:middle line:90% And so this means that the-- 00:23:19.480 --> 00:23:21.640 align:middle line:90% and likewise for the inf. 00:23:21.640 --> 00:23:25.680 align:middle line:84% And so you see that it's this inequality. 00:23:25.680 --> 00:23:30.240 align:middle line:84% You have to, of course, make it nice, into a mathematical proof. 00:23:30.240 --> 00:23:32.650 align:middle line:84% But this inequality will seem kind of obvious. 00:23:32.650 --> 00:23:37.040 align:middle line:90% 00:23:37.040 --> 00:23:41.720 align:middle line:84% Let me instead try and prove that if you 00:23:41.720 --> 00:23:46.920 align:middle line:84% split the integral up into pieces, then 00:23:46.920 --> 00:23:51.080 align:middle line:90% you can just sum the pieces. 00:23:51.080 --> 00:23:52.580 align:middle line:90% So this is the following. 00:23:52.580 --> 00:23:58.960 align:middle line:90% 00:23:58.960 --> 00:24:01.280 align:middle line:84% So again, in the notes, which I haven't-- 00:24:01.280 --> 00:24:03.840 align:middle line:84% I forgot to post, but I will post-- 00:24:03.840 --> 00:24:07.740 align:middle line:84% I guess these are posted because this is from-- 00:24:07.740 --> 00:24:11.346 align:middle line:84% these are things we didn't quite do yet last time. 00:24:11.346 --> 00:24:16.580 align:middle line:84% But the proofs of all of these four here, 00:24:16.580 --> 00:24:19.000 align:middle line:84% also the one in particular-- like the one, 00:24:19.000 --> 00:24:21.620 align:middle line:84% this one here, I didn't do now, they 00:24:21.620 --> 00:24:25.140 align:middle line:84% are all-- the proofs are all in the notes. 00:24:25.140 --> 00:24:30.740 align:middle line:84% So number 4 here is that if you take a function, 00:24:30.740 --> 00:24:35.080 align:middle line:84% and it's integrable on some interval like this, 00:24:35.080 --> 00:24:37.980 align:middle line:90% it's bounded-- always bounded-- 00:24:37.980 --> 00:24:44.660 align:middle line:84% and you're looking at the integral here of this function, 00:24:44.660 --> 00:24:46.980 align:middle line:90% and c here is-- 00:24:46.980 --> 00:24:49.220 align:middle line:90% c is somewhere between. 00:24:49.220 --> 00:24:53.220 align:middle line:90% So c is lying between a and b. 00:24:53.220 --> 00:24:57.400 align:middle line:84% Well, then, actually-- then, actually, 00:24:57.400 --> 00:25:00.660 align:middle line:84% let me just full write down this. 00:25:00.660 --> 00:25:03.260 align:middle line:84% Then the function here-- if you're 00:25:03.260 --> 00:25:06.660 align:middle line:84% think about restricting the function to the smaller 00:25:06.660 --> 00:25:12.660 align:middle line:84% interval from a to c, it's also integrable there. 00:25:12.660 --> 00:25:22.000 align:middle line:84% And likewise, if you restrict it to the integral from c to b, 00:25:22.000 --> 00:25:24.140 align:middle line:90% then it's also integrable here. 00:25:24.140 --> 00:25:29.240 align:middle line:84% So it's integrable on both these subintervals. 00:25:29.240 --> 00:25:37.360 align:middle line:84% And you have that the integral here of f dx 00:25:37.360 --> 00:25:48.400 align:middle line:84% is equal to the sum of the integrals, like that. 00:25:48.400 --> 00:25:52.160 align:middle line:90% 00:25:52.160 --> 00:25:54.050 align:middle line:84% And so now, why is this the case? 00:25:54.050 --> 00:25:59.670 align:middle line:90% 00:25:59.670 --> 00:26:02.630 align:middle line:84% So this is just because, if you take-- 00:26:02.630 --> 00:26:06.600 align:middle line:90% 00:26:06.600 --> 00:26:07.340 align:middle line:90% so let's see. 00:26:07.340 --> 00:26:22.080 align:middle line:90% 00:26:22.080 --> 00:26:22.580 align:middle line:90% Right. 00:26:22.580 --> 00:26:24.480 align:middle line:90% So now, let's try to prove that. 00:26:24.480 --> 00:26:34.420 align:middle line:90% 00:26:34.420 --> 00:26:37.980 align:middle line:90% So right. 00:26:37.980 --> 00:26:40.680 align:middle line:90% So I want to look at the-- 00:26:40.680 --> 00:26:44.860 align:middle line:90% 00:26:44.860 --> 00:26:45.400 align:middle line:90% right. 00:26:45.400 --> 00:26:52.500 align:middle line:84% So let me first prove that if it is integrable here, 00:26:52.500 --> 00:26:55.440 align:middle line:84% it's also integrable on these two subintervals. 00:26:55.440 --> 00:26:58.120 align:middle line:90% And then once we know-- 00:26:58.120 --> 00:27:01.340 align:middle line:90% 00:27:01.340 --> 00:27:08.220 align:middle line:84% in that proof, we will also see what that sum is equal to. 00:27:08.220 --> 00:27:13.520 align:middle line:84% So I again use this lemma from before, that all I need is to-- 00:27:13.520 --> 00:27:17.080 align:middle line:84% for a given epsilon, I need to find a partition 00:27:17.080 --> 00:27:20.360 align:middle line:84% so that the spread between the upper 00:27:20.360 --> 00:27:26.920 align:middle line:84% and the lower sum for each of these two is bounded by epsilon. 00:27:26.920 --> 00:27:35.440 align:middle line:84% So given epsilon, since f here is 00:27:35.440 --> 00:27:41.880 align:middle line:84% integrable on the interval from a to b, 00:27:41.880 --> 00:27:47.400 align:middle line:84% we know that there exists a partition P such 00:27:47.400 --> 00:28:01.030 align:middle line:84% that the upper sum minus the lower sum is less than epsilon. 00:28:01.030 --> 00:28:04.960 align:middle line:90% 00:28:04.960 --> 00:28:07.460 align:middle line:90% Then I look at a refinement. 00:28:07.460 --> 00:28:09.406 align:middle line:90% So here, I have a. 00:28:09.406 --> 00:28:11.820 align:middle line:90% Here, I have b. 00:28:11.820 --> 00:28:13.380 align:middle line:90% And here, I have c. 00:28:13.380 --> 00:28:17.100 align:middle line:90% And then I have some partition. 00:28:17.100 --> 00:28:20.800 align:middle line:84% But the partition may not have c as a dividing point. 00:28:20.800 --> 00:28:40.340 align:middle line:84% So I'm looking at-- so let's let P star be the refinement, that 00:28:40.340 --> 00:29:00.220 align:middle line:84% in addition to all the dividing points of P, 00:29:00.220 --> 00:29:07.510 align:middle line:90% also have c as a dividing point. 00:29:07.510 --> 00:29:14.772 align:middle line:90% 00:29:14.772 --> 00:29:16.820 align:middle line:84% So it has one more dividing point. 00:29:16.820 --> 00:29:19.900 align:middle line:84% I mean, of course, it may be that c was already there. 00:29:19.900 --> 00:29:24.680 align:middle line:84% Then you don't-- then this partition is fine. 00:29:24.680 --> 00:29:27.760 align:middle line:84% And so now-- of course, now the thing 00:29:27.760 --> 00:29:30.820 align:middle line:84% is that once you have that, well, first of all, 00:29:30.820 --> 00:29:32.460 align:middle line:90% you have-- this is refinement. 00:29:32.460 --> 00:29:35.220 align:middle line:84% So you still have-- so you may as well just think-- 00:29:35.220 --> 00:29:36.700 align:middle line:90% so you have this inequality. 00:29:36.700 --> 00:29:47.120 align:middle line:84% So you have that U of f, comma, P star is minus L of f, P star, 00:29:47.120 --> 00:29:50.380 align:middle line:84% is also strictly less than epsilon, right? 00:29:50.380 --> 00:29:53.680 align:middle line:84% That's because it's refinement of P. 00:29:53.680 --> 00:29:59.080 align:middle line:84% And so now you're looking at the Ps, so this thing here. 00:29:59.080 --> 00:30:01.200 align:middle line:90% But this thing here-- 00:30:01.200 --> 00:30:05.140 align:middle line:84% this thing here is just equal to-- 00:30:05.140 --> 00:30:11.860 align:middle line:84% this here is simply equal to U f of P 00:30:11.860 --> 00:30:24.740 align:middle line:84% star intersected with a, c, plus U P, U P 00:30:24.740 --> 00:30:31.200 align:middle line:84% star, intersected with the interval from c to b. 00:30:31.200 --> 00:30:34.620 align:middle line:84% By this here, when I'm taking the partition, 00:30:34.620 --> 00:30:36.800 align:middle line:84% and I intersect it with the interval, 00:30:36.800 --> 00:30:41.740 align:middle line:84% I'm just thinking about the part of the partition that 00:30:41.740 --> 00:30:45.220 align:middle line:90% is dividing this interval up. 00:30:45.220 --> 00:30:46.500 align:middle line:90% So you have this. 00:30:46.500 --> 00:30:52.040 align:middle line:84% And likewise, this thing here is the sum. 00:30:52.040 --> 00:30:57.620 align:middle line:84% This one here is the sum of the partition, the lower sum-- 00:30:57.620 --> 00:31:09.456 align:middle line:84% the sum of the two lower sums, which is this, a to c, plus L f, 00:31:09.456 --> 00:31:17.640 align:middle line:90% P star of c to b, like that. 00:31:17.640 --> 00:31:22.348 align:middle line:90% 00:31:22.348 --> 00:31:25.600 align:middle line:84% And so this, of course, means that-- so now you 00:31:25.600 --> 00:31:26.880 align:middle line:90% can split this up. 00:31:26.880 --> 00:31:29.400 align:middle line:84% And you can divide by-- so you have-- 00:31:29.400 --> 00:31:31.360 align:middle line:90% this is equal to this sum. 00:31:31.360 --> 00:31:33.400 align:middle line:90% This is equal to that sum. 00:31:33.400 --> 00:31:39.280 align:middle line:84% So you can just rearrange it so that this upper sum-- 00:31:39.280 --> 00:31:45.030 align:middle line:84% have. so this means that this whole thing here is equal to U 00:31:45.030 --> 00:31:52.980 align:middle line:84% f, P star, intersected with a, comma, c, minus-- 00:31:52.980 --> 00:31:56.120 align:middle line:84% and then I'll take this one here next-- 00:31:56.120 --> 00:32:05.140 align:middle line:84% so L f, P star, intersected with a, c. 00:32:05.140 --> 00:32:07.480 align:middle line:84% And then I take plus this one here. 00:32:07.480 --> 00:32:16.780 align:middle line:90% 00:32:16.780 --> 00:32:21.200 align:middle line:84% And then I take the last one, which is this one here. 00:32:21.200 --> 00:32:23.820 align:middle line:84% And it, again, comes with a minus sign, 00:32:23.820 --> 00:32:34.740 align:middle line:84% so minus L f, P star, intersected with c, v, 00:32:34.740 --> 00:32:35.620 align:middle line:90% like that. 00:32:35.620 --> 00:32:37.460 align:middle line:90% OK? 00:32:37.460 --> 00:32:39.580 align:middle line:90% But each of these things here-- 00:32:39.580 --> 00:32:42.940 align:middle line:90% this here is non-negative. 00:32:42.940 --> 00:32:46.140 align:middle line:90% This here is non-negative. 00:32:46.140 --> 00:32:51.180 align:middle line:84% And you see, the sum of these two numbers 00:32:51.180 --> 00:32:53.040 align:middle line:90% is bounded by epsilon. 00:32:53.040 --> 00:32:56.720 align:middle line:84% So this means that each of them must be at most epsilon, right? 00:32:56.720 --> 00:32:57.380 align:middle line:90% Yeah? 00:32:57.380 --> 00:32:59.088 align:middle line:84% AUDIENCE: Would you able to explain again 00:32:59.088 --> 00:33:01.920 align:middle line:84% what you did to break up P star into the intersection of a 00:33:01.920 --> 00:33:02.887 align:middle line:90% to c and b to c? 00:33:02.887 --> 00:33:03.470 align:middle line:90% PROFESSOR: OK. 00:33:03.470 --> 00:33:06.400 align:middle line:90% 00:33:06.400 --> 00:33:07.190 align:middle line:90% So I have-- 00:33:07.190 --> 00:33:15.560 align:middle line:90% 00:33:15.560 --> 00:33:17.250 align:middle line:90% I have here the integral. 00:33:17.250 --> 00:33:21.280 align:middle line:90% 00:33:21.280 --> 00:33:27.200 align:middle line:84% I have here the integral from a to b. 00:33:27.200 --> 00:33:30.000 align:middle line:90% Here, I have c. 00:33:30.000 --> 00:33:32.920 align:middle line:84% And the star partition is the one 00:33:32.920 --> 00:33:35.200 align:middle line:84% that has all of the dividing points 00:33:35.200 --> 00:33:39.260 align:middle line:84% of the original partition, P, and then also c. 00:33:39.260 --> 00:33:43.320 align:middle line:84% So the partition looks like this. 00:33:43.320 --> 00:33:46.160 align:middle line:84% Maybe this here would be the partition. 00:33:46.160 --> 00:33:48.780 align:middle line:90% And now I'm thinking about-- 00:33:48.780 --> 00:33:52.800 align:middle line:90% so the sum here in the-- 00:33:52.800 --> 00:33:57.380 align:middle line:84% the sum from the partition, it would look like this. 00:33:57.380 --> 00:34:01.500 align:middle line:84% And I'm summing over all i, right? 00:34:01.500 --> 00:34:04.500 align:middle line:84% But I can look at where I'm summing over just 00:34:04.500 --> 00:34:06.370 align:middle line:90% those intervals, up to here. 00:34:06.370 --> 00:34:08.900 align:middle line:90% 00:34:08.900 --> 00:34:12.340 align:middle line:84% I can look at where I'm just summing over this interval. 00:34:12.340 --> 00:34:15.819 align:middle line:84% And then I can look at where I'm summing over those intervals. 00:34:15.819 --> 00:34:17.520 align:middle line:90% And so that's what I mean. 00:34:17.520 --> 00:34:21.780 align:middle line:84% So this here is now where you're summing over all the i. 00:34:21.780 --> 00:34:24.179 align:middle line:84% But you can also just summing over 00:34:24.179 --> 00:34:28.820 align:middle line:84% the i's that are less or equal to c and the it's that 00:34:28.820 --> 00:34:32.020 align:middle line:90% are in this range, right? 00:34:32.020 --> 00:34:32.730 align:middle line:90% OK, so that's-- 00:34:32.730 --> 00:34:35.586 align:middle line:90% 00:34:35.586 --> 00:34:40.400 align:middle line:90% OK, so this here proves-- 00:34:40.400 --> 00:34:47.100 align:middle line:84% so what this proves is that if you take this function, 00:34:47.100 --> 00:34:50.500 align:middle line:84% and it's integrable on this interval, 00:34:50.500 --> 00:34:53.780 align:middle line:84% and you now restrict it to either of those, 00:34:53.780 --> 00:34:55.800 align:middle line:90% they are also integrable. 00:34:55.800 --> 00:34:57.840 align:middle line:90% That's what it actually proves. 00:34:57.840 --> 00:35:03.120 align:middle line:84% But it's easy to see that it also-- 00:35:03.120 --> 00:35:06.600 align:middle line:84% with a little bit-- maybe I'll leave this for you-- 00:35:06.600 --> 00:35:09.640 align:middle line:84% but that through the proof, we also 00:35:09.640 --> 00:35:12.000 align:middle line:84% get that this sum here will have to be 00:35:12.000 --> 00:35:14.480 align:middle line:90% equal to the sum of these two. 00:35:14.480 --> 00:35:16.600 align:middle line:84% But really, what I, technically speaking, 00:35:16.600 --> 00:35:20.552 align:middle line:90% proved was just this part here. 00:35:20.552 --> 00:35:26.390 align:middle line:84% But again, all of the details for this are in the notes. 00:35:26.390 --> 00:35:30.235 align:middle line:90% 00:35:30.235 --> 00:35:33.360 align:middle line:90% Now, the next thing I want to-- 00:35:33.360 --> 00:35:34.560 align:middle line:90% OK, so yeah. 00:35:34.560 --> 00:35:39.920 align:middle line:84% So I will-- let's talk about one other-- 00:35:39.920 --> 00:35:41.460 align:middle line:90% one corollary of this. 00:35:41.460 --> 00:35:46.000 align:middle line:90% 00:35:46.000 --> 00:35:50.760 align:middle line:84% So it's the corner, really, not of this fourth property, 00:35:50.760 --> 00:35:53.980 align:middle line:90% but on the third property. 00:35:53.980 --> 00:35:57.400 align:middle line:84% So the third property was this with-- 00:35:57.400 --> 00:36:02.940 align:middle line:90% 00:36:02.940 --> 00:36:07.120 align:middle line:84% so the third property was this property here, over here. 00:36:07.120 --> 00:36:09.880 align:middle line:84% So I want to talk about a corollary of this one here. 00:36:09.880 --> 00:36:16.020 align:middle line:90% 00:36:16.020 --> 00:36:18.500 align:middle line:90% So now I have a function. 00:36:18.500 --> 00:36:20.820 align:middle line:84% I'm assuming both the function and the function 00:36:20.820 --> 00:36:22.940 align:middle line:90% that is the absolute value. 00:36:22.940 --> 00:36:27.180 align:middle line:84% So the function that's absolute value means that at each point, 00:36:27.180 --> 00:36:29.140 align:middle line:84% instead of looking at f of x, I'm 00:36:29.140 --> 00:36:32.020 align:middle line:84% looking at the absolute value of f of x. 00:36:32.020 --> 00:36:33.360 align:middle line:90% This is the function. 00:36:33.360 --> 00:36:35.980 align:middle line:84% So I'm assuming that both of these functions 00:36:35.980 --> 00:36:40.320 align:middle line:84% are Riemann integrable on this interval from a to b. 00:36:40.320 --> 00:36:43.060 align:middle line:90% 00:36:43.060 --> 00:36:52.020 align:middle line:84% And then the claim is that the integral here of f is less 00:36:52.020 --> 00:36:55.140 align:middle line:90% or equal-- dx-- 00:36:55.140 --> 00:36:59.100 align:middle line:84% is less or equal to the integral from a. 00:36:59.100 --> 00:37:00.280 align:middle line:90% This is a very-- 00:37:00.280 --> 00:37:06.720 align:middle line:84% this is very simple inequality, but extremely useful. 00:37:06.720 --> 00:37:09.360 align:middle line:90% So this is the claim. 00:37:09.360 --> 00:37:11.760 align:middle line:84% So I'm already assuming that things are-- everything 00:37:11.760 --> 00:37:12.860 align:middle line:90% is Riemann integrable. 00:37:12.860 --> 00:37:16.060 align:middle line:84% So I don't need to prove anything like that. 00:37:16.060 --> 00:37:18.840 align:middle line:84% I just need to establish this inequality. 00:37:18.840 --> 00:37:21.040 align:middle line:84% And so, to do that, I'm just looking 00:37:21.040 --> 00:37:25.780 align:middle line:84% at f, where f is less than or equal to the absolute value. 00:37:25.780 --> 00:37:29.120 align:middle line:84% If f is positive, it's equal to the absolute value. 00:37:29.120 --> 00:37:31.860 align:middle line:84% If it's negative, this here is non-negative anyway. 00:37:31.860 --> 00:37:34.220 align:middle line:90% So this is clear. 00:37:34.220 --> 00:37:35.740 align:middle line:90% This is clearly for all x. 00:37:35.740 --> 00:37:39.520 align:middle line:90% 00:37:39.520 --> 00:37:43.240 align:middle line:84% But it's also clear that if I take minus f, you see, 00:37:43.240 --> 00:37:44.880 align:middle line:90% this is also less than this. 00:37:44.880 --> 00:37:48.300 align:middle line:90% 00:37:48.300 --> 00:37:54.540 align:middle line:84% But now, taken together-- so those two things-- 00:37:54.540 --> 00:37:56.580 align:middle line:90% so now, the first one-- 00:37:56.580 --> 00:38:04.060 align:middle line:84% so let's say star here and double star here. 00:38:04.060 --> 00:38:12.260 align:middle line:84% So star here plus property 3 from over there, 00:38:12.260 --> 00:38:17.980 align:middle line:84% this just gives us that the integral of f from a to b dx 00:38:17.980 --> 00:38:24.140 align:middle line:84% is less or equal to the integral of the absolute value. 00:38:24.140 --> 00:38:27.540 align:middle line:90% This is from star and 3. 00:38:27.540 --> 00:38:35.340 align:middle line:84% But if we take double star here plus 3, 00:38:35.340 --> 00:38:39.200 align:middle line:90% well, then we have that-- 00:38:39.200 --> 00:38:39.840 align:middle line:90% so this here. 00:38:39.840 --> 00:38:44.820 align:middle line:84% So we have that the integral of minus f dx 00:38:44.820 --> 00:38:48.900 align:middle line:84% is less or equal to the integral like this. 00:38:48.900 --> 00:38:53.820 align:middle line:90% 00:38:53.820 --> 00:38:57.240 align:middle line:84% But this one here is-- you can just think about this here 00:38:57.240 --> 00:38:58.660 align:middle line:90% as minus 1. 00:38:58.660 --> 00:39:04.112 align:middle line:84% This is like minus 1 times this function. 00:39:04.112 --> 00:39:06.220 align:middle line:90% So minus 1 just comes out. 00:39:06.220 --> 00:39:07.840 align:middle line:90% That was like number 1. 00:39:07.840 --> 00:39:11.160 align:middle line:84% So this here is like minus 1 times 00:39:11.160 --> 00:39:15.400 align:middle line:84% the integral from a to b of f dx. 00:39:15.400 --> 00:39:18.200 align:middle line:84% But this is, of course, the same as minus the integral 00:39:18.200 --> 00:39:21.420 align:middle line:90% from a to b f dx. 00:39:21.420 --> 00:39:22.860 align:middle line:90% So you see that-- 00:39:22.860 --> 00:39:26.720 align:middle line:84% so this here-- so the second here 00:39:26.720 --> 00:39:33.040 align:middle line:84% implies that minus the integral from a to b f dx 00:39:33.040 --> 00:39:36.180 align:middle line:84% is less or equal to this thing here. 00:39:36.180 --> 00:39:40.172 align:middle line:90% 00:39:40.172 --> 00:39:45.620 align:middle line:84% And so now you have this thing here and that thing here. 00:39:45.620 --> 00:39:49.140 align:middle line:84% And so, together, it just means that the absolute value of this 00:39:49.140 --> 00:39:51.580 align:middle line:90% is bounded by this. 00:39:51.580 --> 00:39:53.330 align:middle line:84% And so that's really this corner. 00:39:53.330 --> 00:39:57.400 align:middle line:90% 00:39:57.400 --> 00:39:59.480 align:middle line:84% The next thing I want to talk about-- 00:39:59.480 --> 00:40:02.060 align:middle line:84% so this is how to evaluate integrals. 00:40:02.060 --> 00:40:05.730 align:middle line:84% And so there, you have the fundamental theorem of calculus. 00:40:05.730 --> 00:40:09.020 align:middle line:90% 00:40:09.020 --> 00:40:12.900 align:middle line:84% And so we'll give two versions of that. 00:40:12.900 --> 00:40:22.540 align:middle line:90% So how to evaluate integrals. 00:40:22.540 --> 00:40:27.940 align:middle line:90% 00:40:27.940 --> 00:40:30.410 align:middle line:84% And so this is the fundamental theorem of calculus. 00:40:30.410 --> 00:40:46.250 align:middle line:90% 00:40:46.250 --> 00:40:48.270 align:middle line:84% And now, what does the fundamental theorem 00:40:48.270 --> 00:40:51.830 align:middle line:84% of calculus-- so let's give version 1. 00:40:51.830 --> 00:41:02.350 align:middle line:84% So version 1-- version 1 is that if you take a-- 00:41:02.350 --> 00:41:15.990 align:middle line:84% if you take a function, f, and f here is continuous, 00:41:15.990 --> 00:41:19.830 align:middle line:84% and you're looking at the integral-- 00:41:19.830 --> 00:41:23.470 align:middle line:84% you're looking at-- so you have a function. 00:41:23.470 --> 00:41:28.950 align:middle line:84% I don't really think about it as two versions. 00:41:28.950 --> 00:41:36.490 align:middle line:84% But the version-- but I divided it into two versions here. 00:41:36.490 --> 00:41:40.030 align:middle line:84% So the function-- if f is continuous, say, 00:41:40.030 --> 00:41:42.810 align:middle line:90% on the interval from-- 00:41:42.810 --> 00:41:47.130 align:middle line:84% on this interval here, and you define a function, f 00:41:47.130 --> 00:41:58.170 align:middle line:84% of x, which is the integral from a to x of f of s ds, 00:41:58.170 --> 00:42:02.170 align:middle line:84% you see, for each fixed x in the interval 00:42:02.170 --> 00:42:05.890 align:middle line:84% here, well, if you're thinking about the function, 00:42:05.890 --> 00:42:10.150 align:middle line:84% little f, little f was continuous on the interval. 00:42:10.150 --> 00:42:12.630 align:middle line:90% So it's continuous on-- 00:42:12.630 --> 00:42:17.690 align:middle line:84% so f here is continuous on the interval from a to x. 00:42:17.690 --> 00:42:22.010 align:middle line:84% So when you restrict it, it's still continuous. 00:42:22.010 --> 00:42:24.690 align:middle line:84% So this means that it's integrable. 00:42:24.690 --> 00:42:28.130 align:middle line:84% And so this here gives you a number. 00:42:28.130 --> 00:42:31.970 align:middle line:84% And so, for each x, you can think about this number here. 00:42:31.970 --> 00:42:34.970 align:middle line:84% So this gives you a function, capital F. 00:42:34.970 --> 00:42:39.150 align:middle line:84% And the claim is that this function here-- 00:42:39.150 --> 00:42:46.190 align:middle line:84% so the claim is that the function capital F, defined 00:42:46.190 --> 00:42:49.570 align:middle line:90% this way, is differentiable. 00:42:49.570 --> 00:42:52.210 align:middle line:90% So capital F is differentiable. 00:42:52.210 --> 00:42:57.310 align:middle line:90% 00:42:57.310 --> 00:43:12.670 align:middle line:84% And the derivative of this at x is little f of x. 00:43:12.670 --> 00:43:13.950 align:middle line:90% And maybe I'll just-- 00:43:13.950 --> 00:43:14.590 align:middle line:90% yeah. 00:43:14.590 --> 00:43:15.900 align:middle line:90% So that's the claim. 00:43:15.900 --> 00:43:19.670 align:middle line:90% 00:43:19.670 --> 00:43:21.710 align:middle line:90% OK. 00:43:21.710 --> 00:43:23.010 align:middle line:90% Let's try to prove that. 00:43:23.010 --> 00:43:30.350 align:middle line:90% 00:43:30.350 --> 00:43:32.870 align:middle line:90% So let's look at-- 00:43:32.870 --> 00:43:34.080 align:middle line:90% let's try to prove-- 00:43:34.080 --> 00:43:37.250 align:middle line:90% 00:43:37.250 --> 00:43:39.370 align:middle line:90% so we want to prove-- 00:43:39.370 --> 00:43:42.700 align:middle line:84% so we want to prove that this function-- so we want to prove-- 00:43:42.700 --> 00:43:45.650 align:middle line:90% 00:43:45.650 --> 00:44:04.170 align:middle line:84% so we want to show that f prime exists at, say, at x0. 00:44:04.170 --> 00:44:07.310 align:middle line:84% If you want to prove that it exists, 00:44:07.310 --> 00:44:10.437 align:middle line:84% then you need to look at the difference quotient. 00:44:10.437 --> 00:44:12.520 align:middle line:84% So we'll need to look at this difference quotient. 00:44:12.520 --> 00:44:16.330 align:middle line:90% 00:44:16.330 --> 00:44:18.570 align:middle line:84% And we need to prove that this thing here 00:44:18.570 --> 00:44:21.350 align:middle line:90% has a limit as x goes to x0. 00:44:21.350 --> 00:44:22.890 align:middle line:90% So x0 is fixed. 00:44:22.890 --> 00:44:25.320 align:middle line:84% And you want to prove that it has a limit, right? 00:44:25.320 --> 00:44:31.115 align:middle line:90% 00:44:31.115 --> 00:44:35.670 align:middle line:84% So in order to estimate this, let me just-- 00:44:35.670 --> 00:44:38.390 align:middle line:90% let me not divide first. 00:44:38.390 --> 00:44:42.430 align:middle line:84% Let me just look at what this difference here is. 00:44:42.430 --> 00:44:46.630 align:middle line:84% And then, afterwards, I'm going to divide by x minus x0. 00:44:46.630 --> 00:44:50.510 align:middle line:84% So if I'm looking at this, let's, for the sake of argument, 00:44:50.510 --> 00:44:54.110 align:middle line:90% assume that x is bigger than x0. 00:44:54.110 --> 00:44:58.470 align:middle line:84% Of course, I need to also see when I'm approaching x0 00:44:58.470 --> 00:45:00.350 align:middle line:90% from the other side. 00:45:00.350 --> 00:45:03.170 align:middle line:84% But the argument for that is similar. 00:45:03.170 --> 00:45:09.270 align:middle line:84% So I just assume that x is bigger than x0. 00:45:09.270 --> 00:45:11.930 align:middle line:90% Well, this difference here-- 00:45:11.930 --> 00:45:15.390 align:middle line:84% well, the function-- so this function here, 00:45:15.390 --> 00:45:17.390 align:middle line:84% the function here-- so let me just write it over 00:45:17.390 --> 00:45:20.910 align:middle line:84% here so I have a little more space. 00:45:20.910 --> 00:45:23.950 align:middle line:84% So this difference here, where the function-- this 00:45:23.950 --> 00:45:30.910 align:middle line:84% here is the integral from a to x f of s ds. 00:45:30.910 --> 00:45:41.210 align:middle line:84% And this thing here is the integral from a to x0 f of s ds, 00:45:41.210 --> 00:45:43.130 align:middle line:90% right? 00:45:43.130 --> 00:45:49.210 align:middle line:84% But this first integral here, I can use this fourth rule 00:45:49.210 --> 00:45:51.190 align:middle line:90% that I maybe don't have anymore. 00:45:51.190 --> 00:45:55.970 align:middle line:84% But the fourth rule is saying that you can split it up. 00:45:55.970 --> 00:45:57.630 align:middle line:84% An integral, you can split it up. 00:45:57.630 --> 00:46:01.690 align:middle line:84% So here, if you have a, then you have x0, 00:46:01.690 --> 00:46:06.690 align:middle line:84% and then you have x, if you want to integrate little f over this, 00:46:06.690 --> 00:46:09.250 align:middle line:84% it's the same as integrating over this 00:46:09.250 --> 00:46:11.770 align:middle line:90% plus the integral over that. 00:46:11.770 --> 00:46:13.070 align:middle line:90% So I do that. 00:46:13.070 --> 00:46:16.210 align:middle line:84% So this one here, I break into two pieces. 00:46:16.210 --> 00:46:21.970 align:middle line:84% I integrate from a to x0 f of s ds. 00:46:21.970 --> 00:46:24.870 align:middle line:84% And then I take the second piece, 00:46:24.870 --> 00:46:32.230 align:middle line:84% which is from x0 to x, and then minus the integral from a 00:46:32.230 --> 00:46:36.190 align:middle line:90% to x0 of f of s ds. 00:46:36.190 --> 00:46:38.550 align:middle line:90% OK? 00:46:38.550 --> 00:46:41.430 align:middle line:84% But now you see that this one here 00:46:41.430 --> 00:46:43.690 align:middle line:90% cancels out with that, right? 00:46:43.690 --> 00:46:51.050 align:middle line:84% And so this difference is just the integral from x0 to x f of s 00:46:51.050 --> 00:46:51.550 align:middle line:90% ds. 00:46:51.550 --> 00:46:59.242 align:middle line:90% 00:46:59.242 --> 00:47:09.190 align:middle line:84% So what we have is we have that this difference here, 00:47:09.190 --> 00:47:13.830 align:middle line:90% f of x minus f of x0-- 00:47:13.830 --> 00:47:19.710 align:middle line:84% we know that this difference here is equal to this integral 00:47:19.710 --> 00:47:20.210 align:middle line:90% here. 00:47:20.210 --> 00:47:24.470 align:middle line:90% 00:47:24.470 --> 00:47:30.690 align:middle line:84% Now, let me use this to bound this thing here from above 00:47:30.690 --> 00:47:31.920 align:middle line:90% and bound it from below. 00:47:31.920 --> 00:47:35.890 align:middle line:90% 00:47:35.890 --> 00:47:39.730 align:middle line:84% You see, this integral here, this function here-- 00:47:39.730 --> 00:47:45.516 align:middle line:84% so this function here, if you're looking at this integral here 00:47:45.516 --> 00:47:50.650 align:middle line:84% of f of s ds, well, this function 00:47:50.650 --> 00:47:57.690 align:middle line:90% here is smaller than the max-- 00:47:57.690 --> 00:48:00.730 align:middle line:84% the function is continuous, so the max and sup is the same-- 00:48:00.730 --> 00:48:05.210 align:middle line:84% the max, or the sup, or whatever you want, on that interval. 00:48:05.210 --> 00:48:14.690 align:middle line:84% So this thing here is less than the integral of, if you will, x0 00:48:14.690 --> 00:48:20.510 align:middle line:84% to x of the sup of f on this interval here. 00:48:20.510 --> 00:48:24.690 align:middle line:90% 00:48:24.690 --> 00:48:27.010 align:middle line:90% But this is just a constant. 00:48:27.010 --> 00:48:29.960 align:middle line:90% And so this here is just-- 00:48:29.960 --> 00:48:33.190 align:middle line:84% this is just like the sup of the function 00:48:33.190 --> 00:48:37.230 align:middle line:90% on this little interval times-- 00:48:37.230 --> 00:48:38.770 align:middle line:90% so this here's a constant. 00:48:38.770 --> 00:48:40.590 align:middle line:90% So this just pops out. 00:48:40.590 --> 00:48:43.210 align:middle line:84% And then it's just the integral of 1 on this. 00:48:43.210 --> 00:48:45.190 align:middle line:84% But that's just the length of the interval. 00:48:45.190 --> 00:48:48.230 align:middle line:84% So this is this here times the length of the interval. 00:48:48.230 --> 00:48:53.096 align:middle line:84% And the length of the interval is this. 00:48:53.096 --> 00:48:58.118 align:middle line:84% So this thing here is less than that. 00:48:58.118 --> 00:49:02.350 align:middle line:84% Likewise, this function here on this little interval 00:49:02.350 --> 00:49:05.370 align:middle line:90% is bigger or equal to the inf. 00:49:05.370 --> 00:49:10.550 align:middle line:84% So this is bigger than if you were just 00:49:10.550 --> 00:49:14.100 align:middle line:84% integrating the inf of the function on this same interval. 00:49:14.100 --> 00:49:17.150 align:middle line:90% 00:49:17.150 --> 00:49:20.750 align:middle line:84% But the inf-- this is just a constant. 00:49:20.750 --> 00:49:22.190 align:middle line:90% This pops out. 00:49:22.190 --> 00:49:26.410 align:middle line:84% So this thing here is just the inf 00:49:26.410 --> 00:49:31.530 align:middle line:84% of the function on this interval here 00:49:31.530 --> 00:49:37.750 align:middle line:84% times the length of the interval, like that. 00:49:37.750 --> 00:49:40.876 align:middle line:90% 00:49:40.876 --> 00:49:44.210 align:middle line:84% So now I can-- and remember that this thing here 00:49:44.210 --> 00:49:46.690 align:middle line:90% was equal to the difference. 00:49:46.690 --> 00:49:48.790 align:middle line:90% So now I can just record this. 00:49:48.790 --> 00:49:52.690 align:middle line:90% So what I have is now-- 00:49:52.690 --> 00:49:58.330 align:middle line:84% what I'm interested in is in the difference in capital F. 00:49:58.330 --> 00:50:01.230 align:middle line:90% So what I'm interested in is-- 00:50:01.230 --> 00:50:04.610 align:middle line:84% of course, I really want to look at the difference quotient. 00:50:04.610 --> 00:50:06.590 align:middle line:84% But I can do this in just a second. 00:50:06.590 --> 00:50:10.930 align:middle line:84% So I'm interested in the difference of the function 00:50:10.930 --> 00:50:13.390 align:middle line:90% at those two points. 00:50:13.390 --> 00:50:18.290 align:middle line:84% I know that that difference is equal to that integral here. 00:50:18.290 --> 00:50:23.130 align:middle line:84% And that integral is bounded from above by this and below 00:50:23.130 --> 00:50:24.010 align:middle line:90% by that. 00:50:24.010 --> 00:50:25.170 align:middle line:90% So I just write-- 00:50:25.170 --> 00:50:28.250 align:middle line:84% I can now just write that this difference here-- 00:50:28.250 --> 00:50:31.030 align:middle line:84% maybe I should write it so I have a little more space. 00:50:31.030 --> 00:50:36.470 align:middle line:84% Somehow, F of x minus F of x0, the difference 00:50:36.470 --> 00:50:42.990 align:middle line:84% is bounded from above by this thing here, the sup of f, 00:50:42.990 --> 00:50:48.430 align:middle line:84% on this little interval, times x minus x0. 00:50:48.430 --> 00:50:55.110 align:middle line:84% And it's bounded from below by the inf 00:50:55.110 --> 00:51:02.430 align:middle line:84% on that same interval times the length of the interval. 00:51:02.430 --> 00:51:06.110 align:middle line:90% 00:51:06.110 --> 00:51:08.630 align:middle line:84% So now we can go to the difference quotient 00:51:08.630 --> 00:51:12.030 align:middle line:84% because we just divide by x minus x0 on both sides. 00:51:12.030 --> 00:51:14.750 align:middle line:90% x is bigger than x0. 00:51:14.750 --> 00:51:17.430 align:middle line:90% So you have this thing here. 00:51:17.430 --> 00:51:19.490 align:middle line:84% So you have this difference quotient here. 00:51:19.490 --> 00:51:22.890 align:middle line:84% And that's what we want to show has a limit. 00:51:22.890 --> 00:51:26.010 align:middle line:84% And now we have that this is less than sup 00:51:26.010 --> 00:51:30.290 align:middle line:84% of f on this little interval here. 00:51:30.290 --> 00:51:34.450 align:middle line:84% And it's bigger or equal to the inf of f 00:51:34.450 --> 00:51:35.920 align:middle line:90% on that same little interval. 00:51:35.920 --> 00:51:39.520 align:middle line:90% 00:51:39.520 --> 00:51:43.650 align:middle line:90% But you see that as x here-- 00:51:43.650 --> 00:51:44.560 align:middle line:90% but now you have-- 00:51:44.560 --> 00:51:48.930 align:middle line:90% 00:51:48.930 --> 00:51:51.680 align:middle line:84% now you have that as x goes to x0-- 00:51:51.680 --> 00:52:02.790 align:middle line:90% 00:52:02.790 --> 00:52:11.130 align:middle line:84% so now, as x goes to x0, the values-- 00:52:11.130 --> 00:52:15.570 align:middle line:84% and not just the value at x, but actually, in this whole interval 00:52:15.570 --> 00:52:16.210 align:middle line:90% here-- 00:52:16.210 --> 00:52:20.970 align:middle line:84% the values goes to what the values is at x0. 00:52:20.970 --> 00:52:26.070 align:middle line:84% So you see that this here, the right-hand side here, goes 00:52:26.070 --> 00:52:29.190 align:middle line:90% to f of x-- little f of x0. 00:52:29.190 --> 00:52:34.430 align:middle line:84% And the left-hand side goes to little f of x0. 00:52:34.430 --> 00:52:37.130 align:middle line:84% So this means that this thing is squeezed between these two 00:52:37.130 --> 00:52:37.630 align:middle line:90% numbers. 00:52:37.630 --> 00:52:39.210 align:middle line:90% That goes to the same number. 00:52:39.210 --> 00:52:42.290 align:middle line:84% So this difference quotient also goes-- 00:52:42.290 --> 00:52:47.950 align:middle line:84% so this difference quotient here goes 00:52:47.950 --> 00:52:54.190 align:middle line:84% to f of x0, because it's squeezed 00:52:54.190 --> 00:52:59.150 align:middle line:84% between those two numbers that converge to the same number-- 00:52:59.150 --> 00:53:03.470 align:middle line:84% namely, to f of x-- little f of x0, right? 00:53:03.470 --> 00:53:08.630 align:middle line:84% So this here proves that it's-- but technically speaking, 00:53:08.630 --> 00:53:13.950 align:middle line:84% I only did it where I'm assuming that x was bigger than x0. 00:53:13.950 --> 00:53:18.570 align:middle line:84% If you let-- so you also need to check 00:53:18.570 --> 00:53:21.290 align:middle line:84% when x is strictly less than x0, and it's 00:53:21.290 --> 00:53:25.010 align:middle line:90% converging to x0 from below. 00:53:25.010 --> 00:53:30.240 align:middle line:84% But with trivial modification, the same proof works. 00:53:30.240 --> 00:53:32.830 align:middle line:84% And that's done out in the notes. 00:53:32.830 --> 00:53:34.920 align:middle line:90% So I will not go through that. 00:53:34.920 --> 00:53:38.430 align:middle line:90% 00:53:38.430 --> 00:53:40.150 align:middle line:90% Now let's talk about-- 00:53:40.150 --> 00:53:43.970 align:middle line:84% so this is the second version of the fundamental theorem 00:53:43.970 --> 00:53:44.500 align:middle line:90% of calculus. 00:53:44.500 --> 00:53:51.810 align:middle line:90% 00:53:51.810 --> 00:53:55.890 align:middle line:84% And so-- again, you wouldn't really 00:53:55.890 --> 00:53:58.250 align:middle line:84% eventually think about it, probably, 00:53:58.250 --> 00:54:00.330 align:middle line:90% as two different versions. 00:54:00.330 --> 00:54:07.410 align:middle line:84% But they're slightly-- it's two things about the same stuff. 00:54:07.410 --> 00:54:09.510 align:middle line:90% But the second version here-- 00:54:09.510 --> 00:54:11.670 align:middle line:84% So I divided it into these two versions. 00:54:11.670 --> 00:54:14.810 align:middle line:90% So the second version-- 00:54:14.810 --> 00:54:26.650 align:middle line:84% so again, fundamental theorem of calculus, version 2. 00:54:26.650 --> 00:54:30.230 align:middle line:90% 00:54:30.230 --> 00:54:32.950 align:middle line:90% And so this is the following-- 00:54:32.950 --> 00:54:37.790 align:middle line:84% that you take a function, F, that is defined-- 00:54:37.790 --> 00:54:46.950 align:middle line:84% and F is defined on some interval from a to b into R. 00:54:46.950 --> 00:54:50.510 align:middle line:90% And F is differentiable. 00:54:50.510 --> 00:54:53.350 align:middle line:90% So this here exists. 00:54:53.350 --> 00:54:57.070 align:middle line:84% It's differentiable at each point. 00:54:57.070 --> 00:55:04.330 align:middle line:84% And f here, this little f, is Riemann integrable. 00:55:04.330 --> 00:55:09.630 align:middle line:90% 00:55:09.630 --> 00:55:19.050 align:middle line:84% And then the claim is that F of b minus capital F at a 00:55:19.050 --> 00:55:28.790 align:middle line:84% is the integral from a to b of little f of x dx. 00:55:28.790 --> 00:55:29.770 align:middle line:90% So that's the claim. 00:55:29.770 --> 00:55:40.422 align:middle line:90% 00:55:40.422 --> 00:55:45.870 align:middle line:84% I wonder if I should just skip that. 00:55:45.870 --> 00:55:50.270 align:middle line:84% I did the proof for this out in the notes also. 00:55:50.270 --> 00:55:52.000 align:middle line:90% So maybe I'll just skip that. 00:55:52.000 --> 00:55:56.730 align:middle line:90% 00:55:56.730 --> 00:56:04.930 align:middle line:84% And then-- OK, so now-- but let me look at an example 00:56:04.930 --> 00:56:06.670 align:middle line:90% where I'm applying it. 00:56:06.670 --> 00:56:19.870 align:middle line:90% 00:56:19.870 --> 00:56:21.710 align:middle line:90% And so I have-- 00:56:21.710 --> 00:56:26.550 align:middle line:84% I'm looking at-- so suppose I want to compute 00:56:26.550 --> 00:56:30.910 align:middle line:90% x squared from 0 to 1 dx. 00:56:30.910 --> 00:56:34.710 align:middle line:90% Suppose I want to compute that. 00:56:34.710 --> 00:56:37.370 align:middle line:84% And so, in this case, I could look at-- 00:56:37.370 --> 00:56:40.350 align:middle line:84% so if I want to apply that version, 00:56:40.350 --> 00:56:42.070 align:middle line:84% I could think about-- suppose I'm 00:56:42.070 --> 00:56:49.256 align:middle line:84% looking at F of x equal to x to the power 3 over 3. 00:56:49.256 --> 00:56:51.590 align:middle line:90% Suppose I'm looking at this. 00:56:51.590 --> 00:56:54.310 align:middle line:84% Well, if I'm looking at this function, 00:56:54.310 --> 00:56:59.630 align:middle line:84% I know that capital F of x is equal to the derivative of this. 00:56:59.630 --> 00:57:01.550 align:middle line:90% It's just x squared. 00:57:01.550 --> 00:57:02.670 align:middle line:90% Right. 00:57:02.670 --> 00:57:05.950 align:middle line:84% So this, say-- and x squared is now-- of course, 00:57:05.950 --> 00:57:07.130 align:middle line:90% it's continuous. 00:57:07.130 --> 00:57:09.110 align:middle line:90% So it's integrable. 00:57:09.110 --> 00:57:13.610 align:middle line:84% So this means that by the second-- 00:57:13.610 --> 00:57:34.610 align:middle line:84% by this second version of the fundamental theorem of calculus, 00:57:34.610 --> 00:57:42.290 align:middle line:84% then we have that F of 1 minus F of 0 00:57:42.290 --> 00:57:47.470 align:middle line:84% is equal to the integral of the derivative, 00:57:47.470 --> 00:57:51.770 align:middle line:84% because the derivative is integrable, so 0 to 1, 00:57:51.770 --> 00:57:57.170 align:middle line:90% of x squared dx. 00:57:57.170 --> 00:58:01.810 align:middle line:84% But this here is now just evaluating this function here. 00:58:01.810 --> 00:58:06.010 align:middle line:90% So this here is 1/3 minus 0. 00:58:06.010 --> 00:58:08.650 align:middle line:84% So this original integral is 1/3. 00:58:08.650 --> 00:58:13.035 align:middle line:90% 00:58:13.035 --> 00:58:13.535 align:middle line:90% OK. 00:58:13.535 --> 00:58:16.750 align:middle line:90% 00:58:16.750 --> 00:58:19.990 align:middle line:90% So the next thing-- 00:58:19.990 --> 00:58:28.330 align:middle line:84% so the next thing I want to talk about is improper integrals. 00:58:28.330 --> 00:58:30.690 align:middle line:84% So this here concludes-- this here 00:58:30.690 --> 00:58:33.230 align:middle line:84% was stuff I actually had from the previous lecture. 00:58:33.230 --> 00:58:35.450 align:middle line:84% That was why I skipped a couple of things. 00:58:35.450 --> 00:58:39.630 align:middle line:90% 00:58:39.630 --> 00:58:43.470 align:middle line:84% But yeah, but the details for the things I skipped 00:58:43.470 --> 00:58:45.010 align:middle line:90% is in the notes. 00:58:45.010 --> 00:58:47.430 align:middle line:84% And the things I skipped is very similar to things 00:58:47.430 --> 00:58:49.040 align:middle line:90% I went through here, anyway. 00:58:49.040 --> 00:58:53.030 align:middle line:90% 00:58:53.030 --> 00:58:56.470 align:middle line:84% So now, let me look at improper integrals. 00:58:56.470 --> 00:59:00.310 align:middle line:84% And there is a couple of versions of this. 00:59:00.310 --> 00:59:02.910 align:middle line:84% So we will look at two versions of this. 00:59:02.910 --> 00:59:05.590 align:middle line:90% One is where the integral-- 00:59:05.590 --> 00:59:07.250 align:middle line:90% where the integral is-- 00:59:07.250 --> 00:59:09.710 align:middle line:84% so we will look at improper integrals. 00:59:09.710 --> 00:59:18.170 align:middle line:90% 00:59:18.170 --> 00:59:21.240 align:middle line:84% So I'm going to look at two versions, so two versions. 00:59:21.240 --> 00:59:28.090 align:middle line:90% 00:59:28.090 --> 00:59:33.870 align:middle line:84% Version 1 is where the integral is unbounded. 00:59:33.870 --> 00:59:39.590 align:middle line:90% The integral is unbounded. 00:59:39.590 --> 00:59:43.490 align:middle line:90% 00:59:43.490 --> 00:59:47.510 align:middle line:84% And the second version is where the function is unbounded. 00:59:47.510 --> 00:59:57.156 align:middle line:90% 00:59:57.156 --> 01:00:02.650 align:middle line:84% Now, in the first version, then you take-- 01:00:02.650 --> 01:00:08.750 align:middle line:90% you're looking at a function, f. 01:00:08.750 --> 01:00:12.310 align:middle line:90% And this function is integrable. 01:00:12.310 --> 01:00:23.010 align:middle line:84% So f here is integrable from a to b for all-- 01:00:23.010 --> 01:00:26.150 align:middle line:90% 01:00:26.150 --> 01:00:29.310 align:middle line:90% so a here is fixed. 01:00:29.310 --> 01:00:34.330 align:middle line:84% But it's integrable for all b bigger than a. 01:00:34.330 --> 01:00:35.410 align:middle line:90% So a is fixed. 01:00:35.410 --> 01:00:36.600 align:middle line:90% So a here, fixed. 01:00:36.600 --> 01:00:39.310 align:middle line:90% 01:00:39.310 --> 01:00:42.710 align:middle line:90% And it's integrable like that. 01:00:42.710 --> 01:00:44.910 align:middle line:90% Then what we want to-- 01:00:44.910 --> 01:00:49.190 align:middle line:90% what we are interested in-- 01:00:49.190 --> 01:00:53.296 align:middle line:84% so again, if you're thinking about the integral as a-- 01:00:53.296 --> 01:00:56.860 align:middle line:84% think about the integral as the weighted area. 01:00:56.860 --> 01:00:59.482 align:middle line:90% 01:00:59.482 --> 01:01:02.150 align:middle line:84% And weight just means that if the function is 01:01:02.150 --> 01:01:05.770 align:middle line:84% negative in a stretch, then the area 01:01:05.770 --> 01:01:08.570 align:middle line:84% there will count with a minus sign. 01:01:08.570 --> 01:01:12.650 align:middle line:84% So this here is-- so what we're interested in is 01:01:12.650 --> 01:01:16.290 align:middle line:84% we were interested in making sense 01:01:16.290 --> 01:01:21.530 align:middle line:84% of the area, the weighted area below the graph, where we 01:01:21.530 --> 01:01:24.810 align:middle line:90% go all the way out to infinity. 01:01:24.810 --> 01:01:26.390 align:middle line:90% And so this is-- 01:01:26.390 --> 01:01:27.520 align:middle line:90% does this make sense? 01:01:27.520 --> 01:01:35.090 align:middle line:90% 01:01:35.090 --> 01:01:38.230 align:middle line:84% So this is, in this case, the improper integral. 01:01:38.230 --> 01:01:41.982 align:middle line:84% It's improper because it's not a bounded interval. 01:01:41.982 --> 01:01:45.290 align:middle line:90% And so yes, if-- 01:01:45.290 --> 01:01:55.330 align:middle line:84% yes, if this limit here of a to b, If this limit here 01:01:55.330 --> 01:02:00.970 align:middle line:84% of f of x dx, if this limit here as b goes to infinity 01:02:00.970 --> 01:02:13.110 align:middle line:84% exists, then we say that the improper integral here 01:02:13.110 --> 01:02:19.310 align:middle line:84% from a to infinity of f of x-- we say that this here exists 01:02:19.310 --> 01:02:21.040 align:middle line:90% and is equal to this limit. 01:02:21.040 --> 01:02:33.780 align:middle line:90% 01:02:33.780 --> 01:02:37.110 align:middle line:84% So this is one version of the improper integrals. 01:02:37.110 --> 01:02:39.550 align:middle line:90% And the other version is-- 01:02:39.550 --> 01:02:42.630 align:middle line:84% the other version of the improper integral 01:02:42.630 --> 01:02:43.650 align:middle line:90% is the following. 01:02:43.650 --> 01:02:51.190 align:middle line:90% 01:02:51.190 --> 01:02:54.270 align:middle line:90% So the second version-- 01:02:54.270 --> 01:02:57.250 align:middle line:84% so this is where f here is unbounded. 01:02:57.250 --> 01:03:00.410 align:middle line:90% 01:03:00.410 --> 01:03:03.530 align:middle line:84% In a minute, I'll look at examples of both things. 01:03:03.530 --> 01:03:05.650 align:middle line:90% So the function is unbounded. 01:03:05.650 --> 01:03:07.210 align:middle line:90% So it's defined. 01:03:07.210 --> 01:03:14.970 align:middle line:84% But f here is supposed to be integrable on, say, a to b. 01:03:14.970 --> 01:03:20.770 align:middle line:84% Right, so a is supposed to be integrable from-- 01:03:20.770 --> 01:03:33.210 align:middle line:84% it's supposed to be integrable from c to b for all c 01:03:33.210 --> 01:03:34.610 align:middle line:90% bigger than-- 01:03:34.610 --> 01:03:40.810 align:middle line:84% so c here, where c is bigger than a and less than b. 01:03:40.810 --> 01:03:44.370 align:middle line:90% So think here about-- 01:03:44.370 --> 01:03:46.810 align:middle line:90% here, you have a. 01:03:46.810 --> 01:03:47.830 align:middle line:90% Here, you have b. 01:03:47.830 --> 01:03:51.010 align:middle line:84% So a and b, in this case, are fixed. 01:03:51.010 --> 01:03:54.330 align:middle line:90% But the function is not bounded. 01:03:54.330 --> 01:03:57.810 align:middle line:84% It's not maybe even defined at a, right? 01:03:57.810 --> 01:03:59.450 align:middle line:90% It's not really defined at a. 01:03:59.450 --> 01:04:02.096 align:middle line:90% 01:04:02.096 --> 01:04:05.750 align:middle line:90% But it is integrable on-- 01:04:05.750 --> 01:04:09.430 align:middle line:84% if you go anywhere strictly to the right of a, 01:04:09.430 --> 01:04:15.550 align:middle line:84% then it is integrable on this interval here. 01:04:15.550 --> 01:04:22.990 align:middle line:84% And so if this limit-- so if this limit here as c goes 01:04:22.990 --> 01:04:29.430 align:middle line:84% to a, the integral from c to b of f of x dx-- 01:04:29.430 --> 01:04:36.110 align:middle line:84% if this limit here exists, then we say that this improper 01:04:36.110 --> 01:04:42.230 align:middle line:84% integral from c to b-- sorry, a to b, of f of x dx-- 01:04:42.230 --> 01:04:45.590 align:middle line:84% this is an improper integral because the function is not 01:04:45.590 --> 01:04:47.870 align:middle line:90% bounded on this whole interval. 01:04:47.870 --> 01:04:50.990 align:middle line:90% It's not even defined at a. 01:04:50.990 --> 01:04:54.030 align:middle line:84% If this here-- we say that this thing here 01:04:54.030 --> 01:05:02.830 align:middle line:84% is equal to this limit as c goes to a of c, b, f of x dx. 01:05:02.830 --> 01:05:05.362 align:middle line:90% 01:05:05.362 --> 01:05:13.970 align:middle line:84% And so the improper integral actually often appear in things. 01:05:13.970 --> 01:05:18.450 align:middle line:84% So let me look at examples of this. 01:05:18.450 --> 01:05:20.590 align:middle line:84% So I will look at example of this. 01:05:20.590 --> 01:05:22.250 align:middle line:84% And I will look-- so again, there's 01:05:22.250 --> 01:05:26.770 align:middle line:84% these two versions, where the interval is unbounded, 01:05:26.770 --> 01:05:29.310 align:middle line:90% or the function is unbounded. 01:05:29.310 --> 01:05:32.030 align:middle line:84% Of course, there's all kinds of variation of this. 01:05:32.030 --> 01:05:34.690 align:middle line:84% Here, I looked at where the function 01:05:34.690 --> 01:05:37.810 align:middle line:90% was unbounded to the left. 01:05:37.810 --> 01:05:40.050 align:middle line:84% But you could do the same to the right. 01:05:40.050 --> 01:05:47.230 align:middle line:84% It could be this interval here, this interval in the-- 01:05:47.230 --> 01:05:50.010 align:middle line:84% where the interval was unbounded. 01:05:50.010 --> 01:05:55.030 align:middle line:84% I'm talking-- I was when I explained what it means, 01:05:55.030 --> 01:06:00.910 align:middle line:84% I did it as if the interval was unbounded to the right. 01:06:00.910 --> 01:06:04.430 align:middle line:84% But of course, you could also look at unbounded to the left, 01:06:04.430 --> 01:06:06.030 align:middle line:84% or you could even look at where it 01:06:06.030 --> 01:06:08.670 align:middle line:84% was unbounded both to the right and to the left. 01:06:08.670 --> 01:06:13.462 align:middle line:90% 01:06:13.462 --> 01:06:15.540 align:middle line:84% But let me look at some examples. 01:06:15.540 --> 01:06:26.630 align:middle line:90% 01:06:26.630 --> 01:06:29.790 align:middle line:90% And so the first example-- 01:06:29.790 --> 01:06:33.670 align:middle line:84% again, the first two examples would be where 01:06:33.670 --> 01:06:37.376 align:middle line:90% the interval is unbounded. 01:06:37.376 --> 01:06:47.670 align:middle line:90% 01:06:47.670 --> 01:06:54.210 align:middle line:84% So this first example here is where the interval is unbounded. 01:06:54.210 --> 01:06:58.970 align:middle line:84% And we're looking at the interval is 1 to infinity. 01:06:58.970 --> 01:07:01.570 align:middle line:84% And we're looking at the function, f 01:07:01.570 --> 01:07:04.680 align:middle line:90% of x equal to 1 over x squared. 01:07:04.680 --> 01:07:09.115 align:middle line:90% 01:07:09.115 --> 01:07:12.370 align:middle line:90% And I'm interested in-- 01:07:12.370 --> 01:07:19.410 align:middle line:84% and the question is, does the improper integral, this here, 01:07:19.410 --> 01:07:23.200 align:middle line:84% of 1 over x squared dx, does this exist? 01:07:23.200 --> 01:07:26.090 align:middle line:90% 01:07:26.090 --> 01:07:29.250 align:middle line:90% That's the question. 01:07:29.250 --> 01:07:35.410 align:middle line:84% And so we use the fundamental theorem of calculus 01:07:35.410 --> 01:07:37.090 align:middle line:90% to determine that. 01:07:37.090 --> 01:07:39.770 align:middle line:84% We observe that if we're looking at f 01:07:39.770 --> 01:07:48.090 align:middle line:84% of x, which is 1 over x, then the derivative here, 01:07:48.090 --> 01:07:51.200 align:middle line:84% of this function here, this is 1 over x squared. 01:07:51.200 --> 01:07:54.526 align:middle line:90% 01:07:54.526 --> 01:07:56.370 align:middle line:90% So this is what we observe. 01:07:56.370 --> 01:08:02.470 align:middle line:84% So therefore, it means that if you're integrating from 1 to b 01:08:02.470 --> 01:08:06.710 align:middle line:84% of 1 over x squared dx, this thing here 01:08:06.710 --> 01:08:11.390 align:middle line:84% is the same as f of b minus f of 1. 01:08:11.390 --> 01:08:14.270 align:middle line:90% 01:08:14.270 --> 01:08:18.149 align:middle line:84% And so, using this, and we know what f is, 01:08:18.149 --> 01:08:19.769 align:middle line:90% you have that this is minus-- 01:08:19.769 --> 01:08:22.790 align:middle line:90% 01:08:22.790 --> 01:08:28.750 align:middle line:90% minus, sorry, 1 over b and-- 01:08:28.750 --> 01:08:29.729 align:middle line:90% no, sorry. 01:08:29.729 --> 01:08:31.410 align:middle line:90% Yeah, it's minus 1 over b. 01:08:31.410 --> 01:08:33.229 align:middle line:90% This is what you get from this. 01:08:33.229 --> 01:08:39.510 align:middle line:84% And here, you're getting 1, plus 1, like this, right? 01:08:39.510 --> 01:08:43.710 align:middle line:84% And so you see that this thing here, as b goes to infinity, 01:08:43.710 --> 01:08:45.630 align:middle line:90% this has converged to 0. 01:08:45.630 --> 01:08:47.830 align:middle line:90% So this has converged to 1. 01:08:47.830 --> 01:08:50.370 align:middle line:90% So indeed, this improper? 01:08:50.370 --> 01:08:54.210 align:middle line:90% Yes is the answer. 01:08:54.210 --> 01:09:03.050 align:middle line:84% And what it's equal to, because it converts to 1, 01:09:03.050 --> 01:09:05.210 align:middle line:90% it's equal to 1. 01:09:05.210 --> 01:09:08.210 align:middle line:84% So again, if you're thinking about it, 01:09:08.210 --> 01:09:09.970 align:middle line:84% it's a little bit like what we did 01:09:09.970 --> 01:09:14.590 align:middle line:84% for series that you're summing infinitely many numbers. 01:09:14.590 --> 01:09:16.609 align:middle line:84% So here, you're looking at something. 01:09:16.609 --> 01:09:19.050 align:middle line:90% And you're looking at the area. 01:09:19.050 --> 01:09:21.210 align:middle line:84% But the area actually is finite, even 01:09:21.210 --> 01:09:22.719 align:middle line:90% though it continues on forever. 01:09:22.719 --> 01:09:26.890 align:middle line:90% 01:09:26.890 --> 01:09:28.630 align:middle line:90% So that's one version of it. 01:09:28.630 --> 01:09:32.390 align:middle line:84% Suppose that instead of 1 over x squared-- 01:09:32.390 --> 01:09:37.130 align:middle line:84% so 1 over the property that the function 1 over x squared has 01:09:37.130 --> 01:09:40.729 align:middle line:84% is that it goes really quite fast to 0. 01:09:40.729 --> 01:09:44.130 align:middle line:84% As x goes to infinity, 1 over x squared 01:09:44.130 --> 01:09:48.350 align:middle line:90% goes really pretty fast to 0. 01:09:48.350 --> 01:09:52.109 align:middle line:84% Suppose I look at a function that does go to 0. 01:09:52.109 --> 01:09:55.590 align:middle line:84% But if it doesn't go to 0, if the function 01:09:55.590 --> 01:09:58.110 align:middle line:90% doesn't go to 0, if we kind of-- 01:09:58.110 --> 01:10:01.630 align:middle line:84% let's say that we're staying above a certain line. 01:10:01.630 --> 01:10:03.310 align:middle line:84% Well, then the area below the graph 01:10:03.310 --> 01:10:05.570 align:middle line:84% would, of course, be larger than this thing. 01:10:05.570 --> 01:10:07.210 align:middle line:90% And that would be infinite. 01:10:07.210 --> 01:10:09.560 align:middle line:90% So it's not going to converge. 01:10:09.560 --> 01:10:15.550 align:middle line:84% So it will have to go to 0 in some way. 01:10:15.550 --> 01:10:18.990 align:middle line:84% So example 2, I'm looking at another function. 01:10:18.990 --> 01:10:20.430 align:middle line:90% Still goes to 0. 01:10:20.430 --> 01:10:24.710 align:middle line:84% Again, the interval is the same from 1 to infinity. 01:10:24.710 --> 01:10:28.910 align:middle line:84% And the function is now-- this time, it's 1 over x. 01:10:28.910 --> 01:10:30.250 align:middle line:90% It still goes to 0. 01:10:30.250 --> 01:10:31.610 align:middle line:90% So maybe it has a chance. 01:10:31.610 --> 01:10:34.310 align:middle line:90% Let's see. 01:10:34.310 --> 01:10:40.390 align:middle line:84% If you now take the integral from 1 to x of-- 01:10:40.390 --> 01:10:47.050 align:middle line:84% 1 to b, sorry, of 1 over x dx, well, again, you 01:10:47.050 --> 01:10:49.230 align:middle line:84% use the fundamental theorem of calculus. 01:10:49.230 --> 01:10:50.730 align:middle line:84% If you're looking at the function, 01:10:50.730 --> 01:11:01.010 align:middle line:84% f of x equal to log of x, the derivative of log is 1 over x. 01:11:01.010 --> 01:11:04.630 align:middle line:84% So this thing here, by the fundamental theorem of calculus, 01:11:04.630 --> 01:11:07.370 align:middle line:84% this second version of fundamental theorem of calculus 01:11:07.370 --> 01:11:11.330 align:middle line:90% is f of b minus f of 1. 01:11:11.330 --> 01:11:17.370 align:middle line:84% And that is log of b minus log of 1. 01:11:17.370 --> 01:11:19.230 align:middle line:90% Log of 1 is 0. 01:11:19.230 --> 01:11:21.370 align:middle line:90% So this is just log of b. 01:11:21.370 --> 01:11:25.570 align:middle line:84% But as b goes to infinity, this here actually goes-- 01:11:25.570 --> 01:11:26.750 align:middle line:90% it's not bounded. 01:11:26.750 --> 01:11:29.450 align:middle line:84% It tends to wander off to infinity. 01:11:29.450 --> 01:11:38.500 align:middle line:84% So this improper integral here, this one here, does not exist. 01:11:38.500 --> 01:11:43.710 align:middle line:90% 01:11:43.710 --> 01:11:46.150 align:middle line:84% It's like the area below this function 01:11:46.150 --> 01:11:51.230 align:middle line:84% here would actually be infinite, right? 01:11:51.230 --> 01:11:53.770 align:middle line:90% It does not go to 0 fast enough. 01:11:53.770 --> 01:12:00.950 align:middle line:84% So that's examples of where the interval is unbounded. 01:12:00.950 --> 01:12:05.870 align:middle line:84% Suppose that we look at examples where the state of the interval 01:12:05.870 --> 01:12:07.250 align:middle line:90% is being unbounded. 01:12:07.250 --> 01:12:09.750 align:middle line:90% The function is unbounded. 01:12:09.750 --> 01:12:11.970 align:middle line:84% So let's look at some examples of that. 01:12:11.970 --> 01:12:14.870 align:middle line:90% 01:12:14.870 --> 01:12:23.150 align:middle line:84% Suppose you take-- so now, suppose you take the interval 01:12:23.150 --> 01:12:24.550 align:middle line:90% from-- 01:12:24.550 --> 01:12:25.990 align:middle line:90% so the interval is bounded. 01:12:25.990 --> 01:12:32.110 align:middle line:84% But the function is not unbounded. 01:12:32.110 --> 01:12:34.310 align:middle line:84% Sorry, the function-- the interval is bounded. 01:12:34.310 --> 01:12:36.710 align:middle line:90% But the function is unbounded. 01:12:36.710 --> 01:12:40.790 align:middle line:84% So I'm looking at, first, f of x equal to 1 01:12:40.790 --> 01:12:43.090 align:middle line:90% over square root of x. 01:12:43.090 --> 01:12:49.270 align:middle line:84% As x goes to 0, square root of x goes to 0. 01:12:49.270 --> 01:12:51.830 align:middle line:84% So this means that 1 over it goes to infinity. 01:12:51.830 --> 01:12:54.570 align:middle line:90% So this function is not bounded. 01:12:54.570 --> 01:12:56.730 align:middle line:84% But if I'm now looking at-- so I want 01:12:56.730 --> 01:12:58.930 align:middle line:84% to look at the improper integral, where 01:12:58.930 --> 01:13:01.050 align:middle line:90% I'm taking this function. 01:13:01.050 --> 01:13:06.170 align:middle line:84% The function is blowing up as I'm going close to 0, 01:13:06.170 --> 01:13:07.730 align:middle line:90% blowing up like that. 01:13:07.730 --> 01:13:12.290 align:middle line:84% And I want to see, does this here-- 01:13:12.290 --> 01:13:17.130 align:middle line:84% does this improper integral here-- 01:13:17.130 --> 01:13:18.060 align:middle line:90% does this exist? 01:13:18.060 --> 01:13:22.127 align:middle line:90% 01:13:22.127 --> 01:13:22.960 align:middle line:90% That's the question. 01:13:22.960 --> 01:13:26.090 align:middle line:90% 01:13:26.090 --> 01:13:29.130 align:middle line:84% And so, in order to figure that out, 01:13:29.130 --> 01:13:33.410 align:middle line:84% I use the fundamental theorem of calculus again. 01:13:33.410 --> 01:13:37.090 align:middle line:90% So if I take the function-- 01:13:37.090 --> 01:13:41.190 align:middle line:84% if I take the function, f of x is 01:13:41.190 --> 01:13:45.120 align:middle line:84% equal to 2 times square root of x, 01:13:45.120 --> 01:13:51.550 align:middle line:84% and I now take the derivative of this, 01:13:51.550 --> 01:13:53.470 align:middle line:84% then I get 1 over square root of x. 01:13:53.470 --> 01:13:56.350 align:middle line:90% 01:13:56.350 --> 01:13:58.830 align:middle line:90% And so this means that-- 01:13:58.830 --> 01:14:01.680 align:middle line:84% so this means that if you now look at-- 01:14:01.680 --> 01:14:12.350 align:middle line:90% 01:14:12.350 --> 01:14:15.670 align:middle line:84% so this means that if you're now looking at the integral 01:14:15.670 --> 01:14:18.830 align:middle line:90% here of-- 01:14:18.830 --> 01:14:22.390 align:middle line:90% 01:14:22.390 --> 01:14:28.390 align:middle line:84% this integral from a, say, to 1 of 1 over square root of x-- 01:14:28.390 --> 01:14:30.910 align:middle line:90% a is strictly positive-- 01:14:30.910 --> 01:14:32.470 align:middle line:90% dx. 01:14:32.470 --> 01:14:42.170 align:middle line:84% This thing here is capital F of 1 minus capital F of a. 01:14:42.170 --> 01:14:45.730 align:middle line:84% And so capital F was this function here. 01:14:45.730 --> 01:14:48.090 align:middle line:90% So this is 2, right? 01:14:48.090 --> 01:14:51.810 align:middle line:84% 2-- square root of 1 plus 1, so that's 2-- 01:14:51.810 --> 01:14:56.810 align:middle line:90% minus 2 square root of a. 01:14:56.810 --> 01:15:01.310 align:middle line:84% But now, as a goes to 0, this here converts to 2. 01:15:01.310 --> 01:15:04.180 align:middle line:84% So this improper integral exists. 01:15:04.180 --> 01:15:04.680 align:middle line:90% So yes. 01:15:04.680 --> 01:15:07.850 align:middle line:90% 01:15:07.850 --> 01:15:15.590 align:middle line:84% This one here exists and is equal to 2. 01:15:15.590 --> 01:15:20.430 align:middle line:90% 01:15:20.430 --> 01:15:26.460 align:middle line:84% Let's look at one last example of this. 01:15:26.460 --> 01:15:46.990 align:middle line:90% 01:15:46.990 --> 01:15:51.870 align:middle line:84% And so this is another example where the function is-- so 01:15:51.870 --> 01:15:58.170 align:middle line:84% again, in this sort of example, the function was unbounded. 01:15:58.170 --> 01:16:00.590 align:middle line:90% It's blowing up. 01:16:00.590 --> 01:16:02.090 align:middle line:90% It's going to infinity. 01:16:02.090 --> 01:16:05.270 align:middle line:84% But the area below the graph-- that's where we calculated-- 01:16:05.270 --> 01:16:08.150 align:middle line:90% is still actually finite. 01:16:08.150 --> 01:16:12.520 align:middle line:84% But what if the function is blowing up faster than this? 01:16:12.520 --> 01:16:18.110 align:middle line:84% So this is an example like this, that if you take-- 01:16:18.110 --> 01:16:21.470 align:middle line:84% it's, again, the interval from 0 to 1. 01:16:21.470 --> 01:16:23.390 align:middle line:84% And the function is now-- now it's 01:16:23.390 --> 01:16:28.830 align:middle line:84% going to infinite faster as x goes to 0. 01:16:28.830 --> 01:16:31.470 align:middle line:90% So this is the function. 01:16:31.470 --> 01:16:35.690 align:middle line:84% And so now we want-- what we're again asking is, 01:16:35.690 --> 01:16:43.056 align:middle line:84% does the improper integral-- so does this improper integral here 01:16:43.056 --> 01:16:43.556 align:middle line:90% exist? 01:16:43.556 --> 01:16:48.050 align:middle line:90% 01:16:48.050 --> 01:16:51.430 align:middle line:84% And let's see if it exists or not. 01:16:51.430 --> 01:16:54.570 align:middle line:84% So we use the fundamental theorem of calculus again. 01:16:54.570 --> 01:16:57.610 align:middle line:84% We observe that the function, F of x 01:16:57.610 --> 01:17:02.770 align:middle line:84% equal to log of x, that the derivative of this function 01:17:02.770 --> 01:17:04.730 align:middle line:90% is just 1 over x. 01:17:04.730 --> 01:17:08.550 align:middle line:84% So this means that if we integrate from a-- a 01:17:08.550 --> 01:17:14.570 align:middle line:84% is now, again, strictly positive, strictly positive. 01:17:14.570 --> 01:17:19.610 align:middle line:84% So if we're looking at this integral here, 01:17:19.610 --> 01:17:22.090 align:middle line:84% this here is-- because of the second version 01:17:22.090 --> 01:17:24.850 align:middle line:84% of the fundamental theorem of calculus, we have 01:17:24.850 --> 01:17:30.230 align:middle line:84% that this here is F in 1 minus F in a. 01:17:30.230 --> 01:17:33.190 align:middle line:90% 01:17:33.190 --> 01:17:39.150 align:middle line:84% And this thing is now log of 1-- that's 0-- 01:17:39.150 --> 01:17:41.670 align:middle line:90% minus log of a. 01:17:41.670 --> 01:17:43.790 align:middle line:90% Log of a is-- 01:17:43.790 --> 01:17:47.910 align:middle line:84% right, so log of a-- so this is like that. 01:17:47.910 --> 01:17:51.670 align:middle line:84% But now, remember that the log function here-- 01:17:51.670 --> 01:17:55.170 align:middle line:84% the log function looks like this. 01:17:55.170 --> 01:17:58.390 align:middle line:90% This is the log function. 01:17:58.390 --> 01:18:02.950 align:middle line:84% So you see that as you go towards 0 here, 01:18:02.950 --> 01:18:05.930 align:middle line:84% this number here goes to minus infinity. 01:18:05.930 --> 01:18:08.630 align:middle line:90% So this here goes to infinity. 01:18:08.630 --> 01:18:11.350 align:middle line:90% So it doesn't converge. 01:18:11.350 --> 01:18:13.850 align:middle line:90% So the answer is no. 01:18:13.850 --> 01:18:19.936 align:middle line:90% 01:18:19.936 --> 01:18:25.670 align:middle line:84% And so the very last thing I want to talk about is, 01:18:25.670 --> 01:18:28.150 align:middle line:90% I'll just give you a definition. 01:18:28.150 --> 01:18:29.720 align:middle line:90% So this is arc length. 01:18:29.720 --> 01:18:33.650 align:middle line:90% 01:18:33.650 --> 01:18:38.200 align:middle line:84% So suppose you take a curve in the plane. 01:18:38.200 --> 01:18:44.690 align:middle line:90% 01:18:44.690 --> 01:18:48.660 align:middle line:84% So suppose that I have a curve in the plane. 01:18:48.660 --> 01:18:56.490 align:middle line:90% 01:18:56.490 --> 01:19:00.290 align:middle line:90% So gamma of t. 01:19:00.290 --> 01:19:01.630 align:middle line:90% This is a curve. 01:19:01.630 --> 01:19:03.370 align:middle line:90% So it has-- in the plane. 01:19:03.370 --> 01:19:07.530 align:middle line:84% So it means that it has two coordinates called gamma of s. 01:19:07.530 --> 01:19:09.220 align:middle line:84% I think that's what I did in the notes. 01:19:09.220 --> 01:19:12.370 align:middle line:90% 01:19:12.370 --> 01:19:15.810 align:middle line:90% So this here is the curve. 01:19:15.810 --> 01:19:22.630 align:middle line:84% And I'm assuming that f and g are differentiable. 01:19:22.630 --> 01:19:25.210 align:middle line:90% 01:19:25.210 --> 01:19:31.130 align:middle line:84% And the derivative here are continuous. 01:19:31.130 --> 01:19:33.870 align:middle line:90% 01:19:33.870 --> 01:19:35.950 align:middle line:90% So suppose I have such a curve. 01:19:35.950 --> 01:19:41.590 align:middle line:84% And s here is running in an interval from a to b. 01:19:41.590 --> 01:19:45.950 align:middle line:84% Then the length of the curve is, by definition, 01:19:45.950 --> 01:19:54.270 align:middle line:84% the integral from a to b of square root of f prime 01:19:54.270 --> 01:20:04.550 align:middle line:84% of s squared plus g prime of s squared ds. 01:20:04.550 --> 01:20:07.530 align:middle line:84% So this is the length of the curve. 01:20:07.530 --> 01:20:11.630 align:middle line:84% Maybe I'll talk a little bit more about that next time. 01:20:11.630 --> 01:20:12.490 align:middle line:90% OK. 01:20:12.490 --> 01:20:14.510 align:middle line:90% So I'll stop here. 01:20:14.510 --> 01:20:19.310 align:middle line:90% And I'll post the notes. 01:20:19.310 --> 01:20:22.350 align:middle line:90% Any questions? 01:20:22.350 --> 01:20:25.060 align:middle line:90% OK, great. 01:20:25.060 --> 01:20:28.000 align:middle line:90%