WEBVTT 00:00:07.680 --> 00:00:08.150 JOEL LEWIS: Hi. 00:00:08.150 --> 00:00:09.690 Welcome back to recitation. 00:00:09.690 --> 00:00:11.600 In lecture, you've been learning about triple 00:00:11.600 --> 00:00:12.520 integration. 00:00:12.520 --> 00:00:15.540 And I have a problem here for you on computing a volume of a 00:00:15.540 --> 00:00:17.830 region using a triple integral. 00:00:17.830 --> 00:00:19.710 So let's look at this. 00:00:19.710 --> 00:00:23.570 So I have a volume and I'm describing it to you; it's the 00:00:23.570 --> 00:00:28.650 volume inside the paraboloid z equals x squared plus y 00:00:28.650 --> 00:00:33.530 squared and bounded by the plane z equals 2y. 00:00:33.530 --> 00:00:35.310 So I've drawn a little picture here for you. 00:00:35.310 --> 00:00:38.770 So this is the paraboloid here. 00:00:38.770 --> 00:00:41.440 And we're just taking a plane cut of it. 00:00:41.440 --> 00:00:44.800 And so this is going to slice off some chunk of that 00:00:44.800 --> 00:00:48.280 paraboloid, and what I want to know is what the volume is of 00:00:48.280 --> 00:00:50.920 that piece that gets cut off by that plane there. 00:00:50.920 --> 00:00:54.730 So below the plane and above the paraboloid. 00:00:54.730 --> 00:00:58.030 So, why don't you pause the video, take some time, work 00:00:58.030 --> 00:00:59.590 out this problem, come back, and we 00:00:59.590 --> 00:01:00.840 can work on it together. 00:01:08.742 --> 00:01:10.940 I hope you had some luck with this problem. 00:01:10.940 --> 00:01:13.350 I think it's a bit of a tricky one, so let's start to work 00:01:13.350 --> 00:01:15.140 through it together. 00:01:15.140 --> 00:01:18.440 So sometimes you have a problem 00:01:18.440 --> 00:01:20.050 with a triple integral. 00:01:20.050 --> 00:01:23.480 And you need to set up your bounds of integration. 00:01:23.480 --> 00:01:25.400 And sometimes you can look at it and it's just 00:01:25.400 --> 00:01:26.510 clear what they are. 00:01:26.510 --> 00:01:30.560 If you're integrating over a cube, life is really easy. 00:01:30.560 --> 00:01:33.600 But in this case, this region that we want to integrate over 00:01:33.600 --> 00:01:36.690 is kind of more complicated to understand. 00:01:36.690 --> 00:01:37.140 Right? 00:01:37.140 --> 00:01:41.070 So it's easy to see-- well, relatively easy to see-- what 00:01:41.070 --> 00:01:43.260 the bounds on z are. 00:01:43.260 --> 00:01:48.500 So let me draw a couple of two-dimensional pictures. 00:01:48.500 --> 00:01:54.450 So I'm going to draw the yz-plane cross section here. 00:01:54.450 --> 00:01:59.680 So in the yz-plane cross section, this paraboloid just 00:01:59.680 --> 00:02:00.680 becomes a parabola. 00:02:00.680 --> 00:02:06.920 So that becomes the parabola z equals y squared, which is a 00:02:06.920 --> 00:02:09.985 plane section of the paraboloid z equals x squared 00:02:09.985 --> 00:02:11.710 plus y squared. 00:02:11.710 --> 00:02:21.250 And this plane z equals 2y becomes the line z equals 2y. 00:02:21.250 --> 00:02:23.580 And this little sliver is a plane section of 00:02:23.580 --> 00:02:25.940 the region in question. 00:02:25.940 --> 00:02:29.970 So we see that z is going from the paraboloid to the plane. 00:02:29.970 --> 00:02:32.500 And over here, we see that z is going from the paraboloid 00:02:32.500 --> 00:02:33.250 to the plane. 00:02:33.250 --> 00:02:36.430 But what we really need to understand then is what the 00:02:36.430 --> 00:02:39.020 relationship between x and y is. 00:02:39.020 --> 00:02:41.490 So what is the shadow of this region? 00:02:41.490 --> 00:02:43.250 How are x and y related to each other? 00:02:43.250 --> 00:02:46.250 How can we bound x in terms of y or y in terms of x? 00:02:46.250 --> 00:02:49.790 Or should we use cylindrical coordinates or what? 00:02:49.790 --> 00:02:51.790 And so in order to that what we need to do is we need to 00:02:51.790 --> 00:02:52.760 figure out-- 00:02:52.760 --> 00:02:55.780 when you project this region down, when you flatten it 00:02:55.780 --> 00:02:58.440 along z, so you're disregarding z now, and then 00:02:58.440 --> 00:02:59.950 you're just looking at its shadow, its 00:02:59.950 --> 00:03:03.400 footprint, in the xy plane-- 00:03:03.400 --> 00:03:05.540 you want to figure out, what is that region? 00:03:05.540 --> 00:03:06.340 What does it look like? 00:03:06.340 --> 00:03:11.460 So somehow we'll project down and there will be some region 00:03:11.460 --> 00:03:14.900 R down here. 00:03:14.900 --> 00:03:22.160 So I'll call this region R. And that region will be the 00:03:22.160 --> 00:03:26.020 projection of this solid region down. 00:03:26.020 --> 00:03:28.740 And it has some boundary curve-- 00:03:28.740 --> 00:03:34.080 C, say-- the boundary curve of the region R. Just in case we 00:03:34.080 --> 00:03:36.570 need to refer to them later, it's good to give them letters 00:03:36.570 --> 00:03:38.410 so that they have names. 00:03:38.410 --> 00:03:41.860 So what we need to figure out now is what is this region R? 00:03:41.860 --> 00:03:47.430 Now this is tough to do by just intuitive reasoning or 00:03:47.430 --> 00:03:49.500 just by looking at this picture I've drawn. 00:03:49.500 --> 00:03:52.860 So in this case, we're kind of forced to use some algebra. 00:03:52.860 --> 00:03:53.360 All right. 00:03:53.360 --> 00:03:57.390 So what do we know about this region R and this curve C? 00:03:57.390 --> 00:04:04.390 Well, C is the projection downwards of the curve of 00:04:04.390 --> 00:04:08.830 intersection of this plane with this paraboloid, right? 00:04:08.830 --> 00:04:12.140 So it's the projection down at this curve intersection. 00:04:12.140 --> 00:04:14.300 So what does that mean about its equation? 00:04:14.300 --> 00:04:19.380 Well, it means it's what we get if we solve for z in one 00:04:19.380 --> 00:04:21.770 of the two equations of the surfaces and 00:04:21.770 --> 00:04:22.960 plug it into the other. 00:04:22.960 --> 00:04:25.490 And that will give us an equation with just x and y, 00:04:25.490 --> 00:04:29.620 and that will be the equation of this curve C. OK. 00:04:29.620 --> 00:04:39.140 So in our case, that means that C is given by this 00:04:39.140 --> 00:04:44.356 equation x squared plus y squared equals 2y. 00:04:44.356 --> 00:04:44.920 All right. 00:04:44.920 --> 00:04:47.940 So whenever x squared plus y squared equals 2y, that's a 00:04:47.940 --> 00:04:53.520 point x, y such that directly above that point is a place 00:04:53.520 --> 00:04:57.020 where the plane intersects the paraboloid. 00:04:57.020 --> 00:04:57.610 Well, OK. 00:04:57.610 --> 00:04:59.180 So what is this curve? 00:04:59.180 --> 00:05:01.910 Well, a little bit of algebra can help us sort that out. 00:05:01.910 --> 00:05:05.380 If you bring the 2y over here and complete the square, you 00:05:05.380 --> 00:05:09.340 can see that we can rewrite this as x squared plus (y 00:05:09.340 --> 00:05:12.530 minus 1) squared equals 1. 00:05:12.530 --> 00:05:16.340 I brought the 2y over, I've added 1 to both sides, and 00:05:16.340 --> 00:05:19.920 I've factored the y part. 00:05:19.920 --> 00:05:23.065 And so this is an easy equation to recognize. 00:05:23.065 --> 00:05:26.180 This is the equation of a circle with center (0, 00:05:26.180 --> 00:05:27.490 1) and radius 1. 00:05:27.490 --> 00:05:29.200 So let's draw that. 00:05:29.200 --> 00:05:33.010 And so here is a picture of what the shadow looks like in 00:05:33.010 --> 00:05:35.790 the xy plane. 00:05:35.790 --> 00:05:42.470 So the center is at height 1, and then it's this circle. 00:05:42.470 --> 00:05:43.315 That's almost a circle. 00:05:43.315 --> 00:05:45.550 It looks enough like a circle for my purposes. 00:05:45.550 --> 00:05:50.840 So this is the region R. It's a circular region of radius 1 00:05:50.840 --> 00:05:53.090 with center (0, 1). 00:05:53.090 --> 00:05:54.090 OK, great. 00:05:54.090 --> 00:05:56.985 So I'm just going to shade that in again because I like 00:05:56.985 --> 00:05:58.350 doing that. 00:05:58.350 --> 00:05:59.100 OK. 00:05:59.100 --> 00:06:01.470 So that's the region R. 00:06:01.470 --> 00:06:02.720 So what is this region R? 00:06:04.760 --> 00:06:05.990 Let's look back over here. 00:06:05.990 --> 00:06:09.860 It's the shadow of our solid region in the xy plane. 00:06:09.860 --> 00:06:11.690 So when you project down, that's the 00:06:11.690 --> 00:06:12.430 region that you get. 00:06:12.430 --> 00:06:15.350 So why do we need that? 00:06:15.350 --> 00:06:18.190 So we know when we set up this triple integral, z is going to 00:06:18.190 --> 00:06:20.605 be going from the paraboloid up to the plane. 00:06:23.400 --> 00:06:25.760 That's going to be the innermost integral, but then 00:06:25.760 --> 00:06:29.340 the middle integral is going to be y in terms of x or x in 00:06:29.340 --> 00:06:30.050 terms of y. 00:06:30.050 --> 00:06:33.550 Or if we do polar coordinates or cylindrical coordinates, 00:06:33.550 --> 00:06:35.490 it's going to be R in terms of theta. 00:06:35.490 --> 00:06:39.890 So we need to figure out what the boundary is, what that 00:06:39.890 --> 00:06:43.680 region looks like over which we'll be integrating for the 00:06:43.680 --> 00:06:46.750 outer two integrals. 00:06:46.750 --> 00:06:49.080 OK, so now I've been saying we could use cylindrical or we 00:06:49.080 --> 00:06:50.010 could use rectangular. 00:06:50.010 --> 00:06:51.020 What do we want to use? 00:06:51.020 --> 00:06:53.640 Well, so this is a circle. 00:06:53.640 --> 00:06:56.860 It's not centered at the origin, but it is tangent to 00:06:56.860 --> 00:07:00.310 one of the axes at the origin. 00:07:00.310 --> 00:07:04.770 So this is a reasonably nice situation to do polar 00:07:04.770 --> 00:07:10.210 coordinates in, or cylindrical coordinates. 00:07:10.210 --> 00:07:12.650 You have to remember from when you learned cylindrical and 00:07:12.650 --> 00:07:16.190 polar coordinates what the equation of such a circle is. 00:07:16.190 --> 00:07:18.930 And so I'm going to write it down here, and I'm going to 00:07:18.930 --> 00:07:22.290 invite you to go look up why this is true 00:07:22.290 --> 00:07:25.520 if you don't remember. 00:07:25.520 --> 00:07:29.330 This curve has an equation in polar-- 00:07:29.330 --> 00:07:31.710 these are the x- and y-axes here-- 00:07:31.710 --> 00:07:37.890 so this curve has in polar coordinates the equation r 00:07:37.890 --> 00:07:42.100 equals 2 sine theta. 00:07:42.100 --> 00:07:42.370 All right. 00:07:42.370 --> 00:07:45.190 So that gives me this curve here. 00:07:45.190 --> 00:07:47.300 The outer boundary. 00:07:47.300 --> 00:07:50.050 And now what I want is, I don't just want the curve. 00:07:50.050 --> 00:07:53.510 I want to integrate over the whole region, and I want to 00:07:53.510 --> 00:07:54.620 integrate over it once. 00:07:54.620 --> 00:07:56.640 Remember, polar coordinates are a little tricky because 00:07:56.640 --> 00:07:59.970 you have to worry about are you overlapping and so on. 00:07:59.970 --> 00:08:01.340 So how does this work? 00:08:01.340 --> 00:08:05.170 At theta equals 0, or at the origin, and then as theta 00:08:05.170 --> 00:08:08.820 grows, we get further and further away. 00:08:08.820 --> 00:08:13.780 So this is our radius growing out. 00:08:13.780 --> 00:08:15.900 And then at pi over 2, we're at the top 00:08:15.900 --> 00:08:16.560 point of the circle. 00:08:16.560 --> 00:08:18.970 And then as it comes back into pi, it comes back in. 00:08:18.970 --> 00:08:23.730 So we want theta going from 0 less than or equal to theta 00:08:23.730 --> 00:08:25.770 less than or equal to pi here. 00:08:25.770 --> 00:08:28.490 So at pi over 2 at the top, and at pi it comes back for 00:08:28.490 --> 00:08:29.360 the first time. 00:08:29.360 --> 00:08:30.810 And what about r? 00:08:30.810 --> 00:08:33.380 Well, it looks like r has to go all the way 00:08:33.380 --> 00:08:34.450 out to 2 sine theta. 00:08:34.450 --> 00:08:36.840 And in fact, we always want it to start at the origin. 00:08:36.840 --> 00:08:41.810 So we always want r to go from 0 to-- 00:08:41.810 --> 00:08:42.690 this outer boundary-- 00:08:42.690 --> 00:08:45.150 2 sine theta. 00:08:45.150 --> 00:08:50.470 So this describes this region big R that we're trying to 00:08:50.470 --> 00:08:50.960 integrate over. 00:08:50.960 --> 00:08:55.210 This circular region in polar coordinates. 00:08:55.210 --> 00:08:55.860 So OK. 00:08:55.860 --> 00:08:57.810 So it's a fairly easy description of polar 00:08:57.810 --> 00:08:58.380 coordinates. 00:08:58.380 --> 00:09:00.700 You could also describe it in rectangular coordinates, and 00:09:00.700 --> 00:09:03.770 you could try to solve the problem that way. 00:09:03.770 --> 00:09:06.440 I'm not going to do it for you, but you could give it a 00:09:06.440 --> 00:09:08.810 shot and see if you can come out with the same answer in 00:09:08.810 --> 00:09:12.050 the end that we do. 00:09:12.050 --> 00:09:13.300 So OK. 00:09:13.300 --> 00:09:14.710 So now, what have we done? 00:09:14.710 --> 00:09:17.700 Well, I haven't written our bounds, so let me write our 00:09:17.700 --> 00:09:19.260 bounds on z right here. 00:09:19.260 --> 00:09:22.580 So we know that z is going from the paraboloid. 00:09:25.120 --> 00:09:27.920 If we look at the paraboloid z equals x 00:09:27.920 --> 00:09:29.470 squared plus y squared-- 00:09:29.470 --> 00:09:32.080 but we're working in cylindrical coordinates now, 00:09:32.080 --> 00:09:34.970 so we need to write this in terms of r and theta-- 00:09:34.970 --> 00:09:40.610 so that's z is going from r squared, and it's going up to 00:09:40.610 --> 00:09:42.920 the plane z equals 2y-- 00:09:42.920 --> 00:09:46.690 now y in cylindrical coordinates is r sine theta. 00:09:46.690 --> 00:09:51.160 So z is going from r squared to 2r sine theta. 00:09:51.160 --> 00:09:52.970 So let's go write that down over here. 00:09:52.970 --> 00:09:56.650 So z is going from-- 00:09:56.650 --> 00:09:58.150 just ignore that-- 00:09:58.150 --> 00:10:03.350 from r squared less than or equal to z, and it's going all 00:10:03.350 --> 00:10:07.560 the way up to 2r sine theta. 00:10:07.560 --> 00:10:12.170 So these three equations describe our region. 00:10:12.170 --> 00:10:13.150 Yeah? 00:10:13.150 --> 00:10:15.850 0 less than theta less than pi: that just says theta. 00:10:15.850 --> 00:10:16.120 OK? 00:10:16.120 --> 00:10:18.400 Then when theta is going from 0 to pi-- 00:10:18.400 --> 00:10:22.600 r going from 0 to 2 sine theta-- that says in the xy 00:10:22.600 --> 00:10:25.410 plane we're tracing out this circular shadow. 00:10:25.410 --> 00:10:28.960 And then as z goes from r squared to 2r sine theta, that 00:10:28.960 --> 00:10:32.380 says above this shadow we're above the paraboloid 00:10:32.380 --> 00:10:33.660 and below the plane. 00:10:33.660 --> 00:10:36.670 So that's exactly the region that we want. 00:10:36.670 --> 00:10:37.130 So OK. 00:10:37.130 --> 00:10:40.080 So now how do we get it to volume after we 00:10:40.080 --> 00:10:40.840 figured this out? 00:10:40.840 --> 00:10:44.580 Well, we write down the triple integral. 00:10:44.580 --> 00:10:50.140 So V, the volume of a region D, is equal to the triple 00:10:50.140 --> 00:10:56.400 integral over that solid of dV. OK? 00:10:56.400 --> 00:11:00.450 And in our case, in cylindrical coordinates, dV is 00:11:00.450 --> 00:11:08.740 going to be dz times r dr d theta, or r dz dr d theta. 00:11:08.740 --> 00:11:09.040 OK? 00:11:09.040 --> 00:11:21.030 So this is equal to, if we're integrating, r dz dr d theta. 00:11:21.030 --> 00:11:23.840 And now we need to put in our bounds. 00:11:23.840 --> 00:11:26.340 If we look over on this side of me, here they are. 00:11:26.340 --> 00:11:28.640 And these are our bounds that we're going to be using. 00:11:28.640 --> 00:11:33.400 So theta is going from 0 to pi. 00:11:33.400 --> 00:11:40.650 And r is going from 0 to 2 sine theta. 00:11:40.650 --> 00:11:48.200 And z is going from r squared to 2r sine theta. 00:11:50.780 --> 00:11:53.680 So this triple integral gives us precisely the 00:11:53.680 --> 00:11:55.720 volume of our region. 00:11:55.720 --> 00:11:58.380 And in order to figure out what that volume is, we just 00:11:58.380 --> 00:12:00.260 have to evaluate this integral. 00:12:00.260 --> 00:12:01.960 So let's start doing that. 00:12:01.960 --> 00:12:05.960 I don't think I'm going to go quite all the way, but I'll do 00:12:05.960 --> 00:12:07.680 most of the work. 00:12:07.680 --> 00:12:08.260 So OK. 00:12:08.260 --> 00:12:10.480 So let's do the innermost integral first. Whenever you 00:12:10.480 --> 00:12:12.120 have a triple integral like this-- a 00:12:12.120 --> 00:12:13.290 nice iterated integral-- 00:12:13.290 --> 00:12:15.670 you always start at the inside and work your way out. 00:12:15.670 --> 00:12:19.010 Well here, our integrand is r, and we're integrating with 00:12:19.010 --> 00:12:21.850 respect to z-- and r doesn't have any z's in it-- 00:12:21.850 --> 00:12:24.690 so this inner integral is going to be easy. 00:12:24.690 --> 00:12:28.340 So I'm going to rewrite this as equal to-- 00:12:28.340 --> 00:12:31.696 we keep our outer two bounds-- so it's still from 0 to pi, 00:12:31.696 --> 00:12:42.980 and it's still from 0 to 2 sine theta, of 2r squared sine 00:12:42.980 --> 00:12:48.460 theta minus r cubed dr d theta. 00:12:48.460 --> 00:12:51.760 So what I've done here is I've just integrated. 00:12:51.760 --> 00:12:56.580 I get the anti-derivative of r dz is rz. 00:12:56.580 --> 00:12:57.890 And so then I take the difference 00:12:57.890 --> 00:12:58.840 between those two bounds. 00:12:58.840 --> 00:13:03.130 So I get r times 2r sine theta minus r times r squared. 00:13:03.130 --> 00:13:07.190 So r times 2r sine theta is 2r squared sine theta. 00:13:07.190 --> 00:13:09.080 Minus r times r squared is minus r cubed. 00:13:09.080 --> 00:13:10.760 OK, so I've just done the first integral. 00:13:10.760 --> 00:13:13.770 So now integrating with respect to r. 00:13:13.770 --> 00:13:16.490 OK, this second one isn't so bad either. 00:13:16.490 --> 00:13:19.240 As far as r is concerned, this is just a polynomial. 00:13:19.240 --> 00:13:22.440 Theta is constant with respect to r when we're doing an 00:13:22.440 --> 00:13:23.320 integral like this. 00:13:23.320 --> 00:13:23.980 So OK. 00:13:23.980 --> 00:13:26.140 So the second integral is not too bad either. 00:13:26.140 --> 00:13:28.620 So this is the integral for our outer 00:13:28.620 --> 00:13:30.450 integral from 0 to pi. 00:13:30.450 --> 00:13:31.210 Sticks around-- 00:13:31.210 --> 00:13:35.350 lets not do this one in one fell swoop I think-- 00:13:35.350 --> 00:13:42.900 so it's going to become 2r cubed over 3, sine theta, 00:13:42.900 --> 00:13:46.360 minus r to the fourth over 4. 00:13:46.360 --> 00:13:51.450 And we're taking that between r equals 0 and r 00:13:51.450 --> 00:13:55.420 equals 2 sine theta. 00:13:55.420 --> 00:13:56.730 And then that whole thing is going to be 00:13:56.730 --> 00:13:59.300 integrated d theta. 00:13:59.300 --> 00:14:01.160 So what do we get when we plug this in? 00:14:01.160 --> 00:14:04.770 Well, at r equals zero, this is just 0, so that's easy. 00:14:04.770 --> 00:14:08.090 And so we need the top one, r equals 2 sine theta. 00:14:08.090 --> 00:14:11.950 So this is going to give me something like 16/3 sine to 00:14:11.950 --> 00:14:16.100 the fourth theta minus 4 sine to the fourth theta, so I 00:14:16.100 --> 00:14:27.950 think that works out to be 4/3 sine to the fourth theta d 00:14:27.950 --> 00:14:30.250 theta, between 0 and pi. 00:14:30.250 --> 00:14:33.950 So now you have to remember how to do integrals like this. 00:14:33.950 --> 00:14:38.190 So this is something you probably learned back in the 00:14:38.190 --> 00:14:44.170 trig integral section of your Calculus I or 18.01 class. 00:14:44.170 --> 00:14:46.540 So when it's an even power here, I think the thing that 00:14:46.540 --> 00:14:50.110 we do is we use our half-angle formulas. 00:14:50.110 --> 00:14:55.040 So now I'm going to tell you what your final steps are. 00:14:55.040 --> 00:14:58.310 So first, you're going to use your half-angle formula. 00:14:58.310 --> 00:15:00.060 So what is that half-angle formula? 00:15:00.060 --> 00:15:08.120 So it's sine squared theta is equal to 1 minus cosine 2 00:15:08.120 --> 00:15:11.410 theta over 2. 00:15:11.410 --> 00:15:13.600 So you're going to have to plug this in here, right? 00:15:13.600 --> 00:15:16.900 Sine to the fourth is sine squared quantity squared. 00:15:16.900 --> 00:15:20.190 And then you're going to have a cosine squared 2 theta, so 00:15:20.190 --> 00:15:22.860 you're going to have to use the double-angle formula. 00:15:22.860 --> 00:15:24.820 This time you're going to have to use the double-angle 00:15:24.820 --> 00:15:27.640 formula for cosine, which is very similar, although not 00:15:27.640 --> 00:15:28.440 exactly the same. 00:15:28.440 --> 00:15:30.570 So you're going to have to use those two 00:15:30.570 --> 00:15:31.920 double-angle formulas. 00:15:31.920 --> 00:15:33.210 After that, you'll have something that is 00:15:33.210 --> 00:15:35.980 straightforward to integrate. 00:15:35.980 --> 00:15:37.290 So you'll have something that's 00:15:37.290 --> 00:15:38.090 straightforward to integrate. 00:15:38.090 --> 00:15:43.090 You'll integrate it, and if I'm not mistaken, what you get 00:15:43.090 --> 00:15:48.400 at the end is that you just get a fairly nice and simple 00:15:48.400 --> 00:15:51.040 pi over 2 as your answer. 00:15:51.040 --> 00:15:54.300 So you can check your work there, and make sure that 00:15:54.300 --> 00:15:56.210 you've got out pi over 2 at the end. 00:15:56.210 --> 00:15:59.970 And hopefully, if you tried to do this using rectangular 00:15:59.970 --> 00:16:02.270 coordinates, you also came out with something 00:16:02.270 --> 00:16:03.040 like this as well. 00:16:03.040 --> 00:16:05.530 In that case, you would have to do a trig substitution at 00:16:05.530 --> 00:16:08.630 some point to evaluate your intervals, or you might have 00:16:08.630 --> 00:16:10.330 an arcsine involved. 00:16:10.330 --> 00:16:13.230 Something like that will happen. 00:16:13.230 --> 00:16:15.380 But it should also give you pi over 2, of course. 00:16:15.380 --> 00:16:17.900 Because it's the same region, just described 00:16:17.900 --> 00:16:18.970 in a different way. 00:16:18.970 --> 00:16:21.930 So let me quickly recap what we did. 00:16:21.930 --> 00:16:24.610 Way back over here, we had this 00:16:24.610 --> 00:16:27.200 description of this region. 00:16:27.200 --> 00:16:32.310 So it was the region above our paraboloid and below a plane. 00:16:32.310 --> 00:16:35.910 And so when we're setting this up, we have to figure out in 00:16:35.910 --> 00:16:38.520 order to do a triple integral over this region in order to 00:16:38.520 --> 00:16:43.440 find its volume, we have to pick an order of integration, 00:16:43.440 --> 00:16:47.900 and then we have to know what the bounds are for the inside 00:16:47.900 --> 00:16:50.170 in terms of the outer two variables, for the middle one 00:16:50.170 --> 00:16:52.370 in terms of the outermost one, and so on. 00:16:52.370 --> 00:17:00.310 So in this case, that means it was a natural choice to make z 00:17:00.310 --> 00:17:02.800 the first variable-- the innermost variable. 00:17:02.800 --> 00:17:05.310 And so then after that, we needed to project to find the 00:17:05.310 --> 00:17:09.580 relationship in the xy plane between the other variables. 00:17:09.580 --> 00:17:17.230 Now in this case, we did that by solving this little algebra 00:17:17.230 --> 00:17:17.780 problem here. 00:17:17.780 --> 00:17:23.280 We solved for z in the two surfaces that we were given, 00:17:23.280 --> 00:17:24.650 and we set them equal to each other. 00:17:24.650 --> 00:17:27.080 And so this gives us a description for the boundary 00:17:27.080 --> 00:17:28.310 curve for our region. 00:17:28.310 --> 00:17:32.690 And because it's a nice circle, this suggested that 00:17:32.690 --> 00:17:34.630 one possibility was cylindrical coordinates. 00:17:34.630 --> 00:17:38.210 So we went ahead, and we found in cylindrical coordinates the 00:17:38.210 --> 00:17:39.840 description of this shadow. 00:17:39.840 --> 00:17:44.090 And then we used the knowledge we previously had to describe 00:17:44.090 --> 00:17:46.140 the whole region in cylindrical coordinates. 00:17:46.140 --> 00:17:48.550 So we had this description of our entire region. 00:17:48.550 --> 00:17:51.030 And then to compute its volume, we just set up the 00:17:51.030 --> 00:17:55.330 triple integral volume is equal to a triple integral dV. 00:17:55.330 --> 00:17:56.310 In our case, dV-- 00:17:56.310 --> 00:17:57.720 since we're in cylindrical coordinates-- 00:17:57.720 --> 00:17:59.890 that's r dz dr d theta. 00:17:59.890 --> 00:18:04.240 We put in our bounds, and then we evaluated the integral. 00:18:04.240 --> 00:18:05.960 I'll stop there.