WEBVTT
00:00:06.866 --> 00:00:07.490
JOEL LEWIS: Hi.
00:00:07.490 --> 00:00:08.970
Welcome back to recitation.
00:00:08.970 --> 00:00:11.150
In lecture, you've been
learning about computing
00:00:11.150 --> 00:00:13.240
double integrals
and about changing
00:00:13.240 --> 00:00:14.360
the order of integration.
00:00:14.360 --> 00:00:16.555
And how you can look
at a given region
00:00:16.555 --> 00:00:19.470
and you can integrate over
it by integrating dx dy
00:00:19.470 --> 00:00:21.380
or by integrating dy dx.
00:00:21.380 --> 00:00:23.470
So here I have some examples.
00:00:23.470 --> 00:00:25.560
I have two regions.
00:00:25.560 --> 00:00:27.930
So one region is
the triangle whose
00:00:27.930 --> 00:00:30.840
vertices are the origin,
the point (0, 2),
00:00:30.840 --> 00:00:33.120
and the point minus 1, 2.
00:00:33.120 --> 00:00:35.530
And the other one is
a sector of a circle.
00:00:35.530 --> 00:00:38.690
So the circle has a radius 2
and is centered at the origin.
00:00:38.690 --> 00:00:42.290
And I want the part of that
circle that's above the x-axis
00:00:42.290 --> 00:00:44.860
and below the line y equals x.
00:00:44.860 --> 00:00:46.450
So what I'd like
you to do is I'd
00:00:46.450 --> 00:00:48.740
like you to write down
what a double integral
00:00:48.740 --> 00:00:51.000
over these regions
looks like, but I'd
00:00:51.000 --> 00:00:52.550
like you to do it
two different ways.
00:00:52.550 --> 00:00:55.750
I'd like you to do it as an
iterated integral in the order
00:00:55.750 --> 00:00:57.610
dx dy.
00:00:57.610 --> 00:01:00.590
And I'd also like you to do
it as an iterated integral
00:01:00.590 --> 00:01:01.890
in the order dy dx.
00:01:01.890 --> 00:01:03.880
So I'd like you to
express the integrals
00:01:03.880 --> 00:01:06.930
over these regions in
terms of iterated integrals
00:01:06.930 --> 00:01:08.860
in both possible orders.
00:01:08.860 --> 00:01:11.371
So why don't you pause the
video, have a go at that,
00:01:11.371 --> 00:01:13.120
come back, and we can
work on it together.
00:01:21.486 --> 00:01:22.860
So the first thing
to do whenever
00:01:22.860 --> 00:01:24.430
you're given a
problem like this--
00:01:24.430 --> 00:01:28.620
and in fact, almost anytime you
have to do a double integral--
00:01:28.620 --> 00:01:30.620
is to try and understand
the region in question.
00:01:30.620 --> 00:01:32.360
It's always a good
idea to understand
00:01:32.360 --> 00:01:33.450
the region in question.
00:01:33.450 --> 00:01:35.570
And by understand the
region in question,
00:01:35.570 --> 00:01:38.011
really the first thing that
I mean is draw a picture.
00:01:38.011 --> 00:01:38.510
All right.
00:01:38.510 --> 00:01:41.580
So let's do part a first.
00:01:41.580 --> 00:01:46.160
So in part a, you
have a triangle,
00:01:46.160 --> 00:01:50.820
it has vertices at the
origin, at the point (0, 2),
00:01:50.820 --> 00:01:53.070
and at the point minus 1, 2.
00:01:56.180 --> 00:02:00.620
So this triangle is
our region in question.
00:02:00.620 --> 00:02:03.150
So now that we've got a
picture of it, we can talk
00:02:03.150 --> 00:02:06.210
and we can say, what
are the boundaries
00:02:06.210 --> 00:02:07.307
of this region, right?
00:02:07.307 --> 00:02:09.140
And we want to know
what its boundaries are.
00:02:09.140 --> 00:02:14.190
So the top boundary is
the line y equals 2,
00:02:14.190 --> 00:02:17.610
the right boundary is
the line x equals 0,
00:02:17.610 --> 00:02:21.260
and this sort of lower left
boundary-- the slanted line--
00:02:21.260 --> 00:02:25.620
is the line y equals minus 2x.
00:02:25.620 --> 00:02:26.120
OK.
00:02:26.120 --> 00:02:30.480
So those are the boundary
edges of this triangle.
00:02:30.480 --> 00:02:33.017
And so now what we
want to figure out
00:02:33.017 --> 00:02:35.350
is we want to figure out, OK,
if you're integrating this
00:02:35.350 --> 00:02:37.910
with respect to x and then y,
or if you're integrating this
00:02:37.910 --> 00:02:39.930
with respect to y
and then x, what
00:02:39.930 --> 00:02:41.890
does that integral look
like when you set it up
00:02:41.890 --> 00:02:43.080
as a double integral.
00:02:43.080 --> 00:02:46.184
So let's start on one of them.
00:02:46.184 --> 00:02:47.350
It doesn't matter which one.
00:02:47.350 --> 00:02:53.520
So let's try and write
the double integral
00:02:53.520 --> 00:02:56.830
over this region R
in the order dx dy.
00:02:56.830 --> 00:03:04.010
OK, so we have
inside bounds dx dy.
00:03:04.010 --> 00:03:05.030
So OK.
00:03:05.030 --> 00:03:07.700
So we need to find
the bounds on x first,
00:03:07.700 --> 00:03:11.540
and those bounds are
going to be in terms of y.
00:03:11.540 --> 00:03:12.800
So the bounds on x.
00:03:12.800 --> 00:03:14.790
So that means when we
look at this region, what
00:03:14.790 --> 00:03:19.250
we want to figure out is we want
to figure out for a given value
00:03:19.250 --> 00:03:23.302
y, what is the
leftmost point and what
00:03:23.302 --> 00:03:24.260
is the rightmost point?
00:03:24.260 --> 00:03:25.520
What are the bounds on x?
00:03:25.520 --> 00:03:32.070
So for given value
y, the largest value
00:03:32.070 --> 00:03:36.740
x is going to take is
along this line x equals 0.
00:03:36.740 --> 00:03:40.430
When you fix some value
of y, the rightmost point
00:03:40.430 --> 00:03:45.710
that x can reach in this region
is at this line x equals 0.
00:03:45.710 --> 00:03:48.660
So x is going to go up to 0.
00:03:48.660 --> 00:03:51.230
That's going to be
its upper bound.
00:03:51.230 --> 00:03:54.980
The lower bound is going to be
the left edge of our region.
00:03:57.890 --> 00:04:01.860
For a given value of y, what is
that leftmost boundary value?
00:04:01.860 --> 00:04:04.030
So what we want to
do is we want to take
00:04:04.030 --> 00:04:07.310
that equation for that boundary
and we want to solve it
00:04:07.310 --> 00:04:09.780
for x in terms of y.
00:04:09.780 --> 00:04:11.750
So that's not hard
to do in this case.
00:04:11.750 --> 00:04:13.860
The line y equals
minus 2x is also
00:04:13.860 --> 00:04:17.060
the line x equals minus 1/2 y.
00:04:17.060 --> 00:04:22.690
So that's that left
boundary: minus 1/2 y.
00:04:22.690 --> 00:04:23.320
OK?
00:04:23.320 --> 00:04:25.680
So then our outer bounds are dy.
00:04:25.680 --> 00:04:28.360
So we want to find the
absolute bounds on y.
00:04:28.360 --> 00:04:30.121
What's the smallest
value that y takes,
00:04:30.121 --> 00:04:31.870
and what's the largest
value that y takes?
00:04:31.870 --> 00:04:34.782
So that means what's the
lowest point of this region
00:04:34.782 --> 00:04:35.740
and what's the highest?
00:04:35.740 --> 00:04:38.000
And so the lowest point
here is the origin.
00:04:38.000 --> 00:04:41.160
So that's when y
takes the value of 0.
00:04:41.160 --> 00:04:43.390
And the highest point-- the
very top of this region--
00:04:43.390 --> 00:04:45.790
is when y equals 2.
00:04:45.790 --> 00:04:46.900
OK.
00:04:46.900 --> 00:04:49.390
So this is what
that double integral
00:04:49.390 --> 00:04:54.460
is going to become when we
evaluate it in the order dx dy.
00:04:54.460 --> 00:04:56.544
So now let's talk
about evaluating it
00:04:56.544 --> 00:04:57.460
in the opposite order.
00:04:57.460 --> 00:05:01.310
So let's switch our
bounds for dy dx.
00:05:01.310 --> 00:05:07.270
So we want the double
integral over R, dy dx.
00:05:07.270 --> 00:05:11.410
OK, so this is going to
be an iterated integral.
00:05:11.410 --> 00:05:14.150
And this time the
inner bounds are
00:05:14.150 --> 00:05:17.000
going to be for y in terms
of x, and the outer bounds
00:05:17.000 --> 00:05:19.030
are going to be
absolute bounds on x.
00:05:19.030 --> 00:05:21.050
So for y in terms
of x, that means we
00:05:21.050 --> 00:05:25.480
look at this region-- we want
to know, for a fixed value of x,
00:05:25.480 --> 00:05:27.860
what's the bottom
boundary of this region,
00:05:27.860 --> 00:05:29.820
and what's the top boundary?
00:05:29.820 --> 00:05:32.830
So here, it's easy to see
that the bottom boundary is
00:05:32.830 --> 00:05:35.840
this line y equals minus
2x, and the top boundary
00:05:35.840 --> 00:05:37.810
is this line y equals 2.
00:05:37.810 --> 00:05:42.310
So y is going from
minus 2x to 2.
00:05:42.310 --> 00:05:42.840
Yeah?
00:05:42.840 --> 00:05:49.355
So for a fixed value of x,
the values of y that give you
00:05:49.355 --> 00:05:51.330
a point in this
region are the values
00:05:51.330 --> 00:05:54.205
that y is at least
minus 2x and at most 2.
00:05:54.205 --> 00:05:55.840
So OK.
00:05:55.840 --> 00:05:57.510
And now we need
the outer bounds.
00:05:57.510 --> 00:06:00.900
So the outer bounds have
to be some real numbers,
00:06:00.900 --> 00:06:02.740
Those are the
absolute bounds on x.
00:06:02.740 --> 00:06:04.830
So we need to know what
the absolute leftmost
00:06:04.830 --> 00:06:07.490
point and the absolute rightmost
point in this region are.
00:06:07.490 --> 00:06:11.510
And so the absolute leftmost
point is this point minus 1, 2.
00:06:11.510 --> 00:06:14.040
So that has an
x-value of minus 1.
00:06:14.040 --> 00:06:16.960
And the absolute
rightmost point is along
00:06:16.960 --> 00:06:20.280
this right edge at x equals 0.
00:06:20.280 --> 00:06:20.970
OK.
00:06:20.970 --> 00:06:25.910
So here are the two integrals.
00:06:25.910 --> 00:06:28.670
The double integral with
respect to x then y,
00:06:28.670 --> 00:06:33.870
and the double integral with
respect to y and then x.
00:06:33.870 --> 00:06:34.400
OK.
00:06:34.400 --> 00:06:36.450
So that's the answer to part a.
00:06:36.450 --> 00:06:39.730
Let's go on to part b.
00:06:39.730 --> 00:06:47.540
So for part b, our region is
we take a circle of radius 2,
00:06:47.540 --> 00:06:51.690
and we take the line y
equals x, and we take
00:06:51.690 --> 00:06:55.980
the line that's the x-axis.
00:06:55.980 --> 00:07:00.830
And so we want a circle,
and we want this sector
00:07:00.830 --> 00:07:02.852
of the circle in here.
00:07:02.852 --> 00:07:09.390
So this region inside the
circle, below the line y
00:07:09.390 --> 00:07:11.590
equal x, and above the x-axis.
00:07:11.590 --> 00:07:13.311
So this wedge of this circle.
00:07:13.311 --> 00:07:13.810
Let's see.
00:07:13.810 --> 00:07:19.420
This value is at x equals
2, this is the origin,
00:07:19.420 --> 00:07:22.212
and this is the
point square root
00:07:22.212 --> 00:07:24.830
of 2 comma square root of 2.
00:07:24.830 --> 00:07:28.430
That common point of the line
y equals x and the circle
00:07:28.430 --> 00:07:30.832
x squared plus y
squared equals 4.
00:07:30.832 --> 00:07:32.290
That's what this
boundary curve is:
00:07:32.290 --> 00:07:35.930
x squared plus y
squared equals 4.
00:07:35.930 --> 00:07:39.310
And of course, this boundary
curve is the line y equals x.
00:07:39.310 --> 00:07:42.450
And this boundary line
is the x-axis, which
00:07:42.450 --> 00:07:44.910
has the equation y equals 0.
00:07:44.910 --> 00:07:46.910
So those are our boundary
curves for our region.
00:07:46.910 --> 00:07:48.740
We've got this nice
picture, so now we
00:07:48.740 --> 00:07:56.110
can talk about expressing it
as an iterated integral in two
00:07:56.110 --> 00:07:56.900
different orders.
00:07:56.900 --> 00:08:01.140
So let's again start off with
this with respect to x first,
00:08:01.140 --> 00:08:02.710
and then with respect to y.
00:08:02.710 --> 00:08:08.000
So we want the double
integral over R, dx dy.
00:08:08.000 --> 00:08:15.360
So this should be an iterated
integral, something dx and then
00:08:15.360 --> 00:08:16.280
dy.
00:08:16.280 --> 00:08:18.340
OK, so we need
bounds on x, which
00:08:18.340 --> 00:08:20.050
means for a fixed
value of y, we need
00:08:20.050 --> 00:08:22.040
to know what is the
leftmost boundary
00:08:22.040 --> 00:08:23.660
and what's the rightmost bound.
00:08:23.660 --> 00:08:27.940
So for a fixed
value of y, we want
00:08:27.940 --> 00:08:30.665
to know what the left edge
is and the right edge is.
00:08:30.665 --> 00:08:32.960
And it's easy to see because
we've drawn this picture,
00:08:32.960 --> 00:08:33.460
right?
00:08:33.460 --> 00:08:36.750
Drawing the picture makes
this a much easier process.
00:08:36.750 --> 00:08:40.275
The left edge is this line y
equals x and the right edge
00:08:40.275 --> 00:08:41.960
is our actual circle.
00:08:41.960 --> 00:08:42.460
Yeah?
00:08:42.460 --> 00:08:45.550
So those are the left
and right boundaries,
00:08:45.550 --> 00:08:48.230
so what we put here are just
the equations of that left edge
00:08:48.230 --> 00:08:52.100
and the equation
of that right edge.
00:08:52.100 --> 00:08:56.840
But we want their equations in
the form x equals something.
00:08:56.840 --> 00:08:58.910
And that's the something
that we put there.
00:08:58.910 --> 00:09:04.750
So for this left edge,
it's the line x equals y.
00:09:04.750 --> 00:09:08.790
So the left bound is y there.
00:09:08.790 --> 00:09:11.810
In this region, x is at least y.
00:09:11.810 --> 00:09:13.696
And the upper bound
here, which is
00:09:13.696 --> 00:09:15.570
going to be the rightmost
bound-- the largest
00:09:15.570 --> 00:09:18.220
value that x takes--
is when x squared
00:09:18.220 --> 00:09:19.820
plus y squared equals 4.
00:09:19.820 --> 00:09:23.350
So when x is equal to the square
root of 4 minus y squared.
00:09:27.030 --> 00:09:28.880
Now you might say
to me, why do I
00:09:28.880 --> 00:09:31.310
know that it's the
positive square root here
00:09:31.310 --> 00:09:32.810
and not the negative
square root?
00:09:32.810 --> 00:09:34.910
And if you said
that to yourself,
00:09:34.910 --> 00:09:35.910
that's a great question.
00:09:35.910 --> 00:09:38.860
And the answer is that
this part of the circle
00:09:38.860 --> 00:09:41.140
is the top half of the
circle and it's also
00:09:41.140 --> 00:09:42.600
the right half of the circle.
00:09:42.600 --> 00:09:45.010
So here we have
positive values of x.
00:09:45.010 --> 00:09:46.960
So it's the right
half of the circle.
00:09:46.960 --> 00:09:49.470
We want the positive
values of x, so we
00:09:49.470 --> 00:09:51.580
want the positive square root.
00:09:51.580 --> 00:09:52.710
OK.
00:09:52.710 --> 00:09:53.210
Good.
00:09:53.210 --> 00:09:55.520
And so those are
the bounds on x.
00:09:55.520 --> 00:09:57.760
Now we need the bounds on y.
00:09:57.760 --> 00:09:59.690
So the bounds on y,
well, what are they?
00:09:59.690 --> 00:10:01.730
Well, we want the
absolute bounds on y. y
00:10:01.730 --> 00:10:04.800
is the outermost variable that
we're integrating with respect
00:10:04.800 --> 00:10:09.270
to, so we want the absolute
bounds-- the absolute lowest
00:10:09.270 --> 00:10:10.910
value that y takes
in this region,
00:10:10.910 --> 00:10:13.090
and the absolute largest
value that y takes.
00:10:13.090 --> 00:10:14.710
So the smallest
value that y takes
00:10:14.710 --> 00:10:17.430
in this region-- that's
the lowest point-- that's
00:10:17.430 --> 00:10:20.160
along this line, and
that's when y equals 0.
00:10:20.160 --> 00:10:23.610
And the largest
value that y takes--
00:10:23.610 --> 00:10:25.830
that's when y is as
large as possible
00:10:25.830 --> 00:10:27.430
as it can get in
this region-- is
00:10:27.430 --> 00:10:29.210
up at this point of
intersection there,
00:10:29.210 --> 00:10:33.460
so that's when y is equal
to the square root of 2.
00:10:33.460 --> 00:10:35.020
OK, three quarters done.
00:10:35.020 --> 00:10:36.940
Yeah?
00:10:36.940 --> 00:10:39.290
This is that iterated integral.
00:10:39.290 --> 00:10:42.902
So now, we want to
do the same thing.
00:10:42.902 --> 00:10:49.800
R-- the integral over
this region R-- dy dx.
00:10:49.800 --> 00:10:50.599
OK.
00:10:50.599 --> 00:10:52.140
So we're going to
look at this region
00:10:52.140 --> 00:10:55.536
and we want to say-- dy is
going to be on the inside--
00:10:55.536 --> 00:10:57.910
so we're going to say, OK, so
we need to know for a fixed
00:10:57.910 --> 00:11:01.395
value of x, what's the
smallest value that y can take
00:11:01.395 --> 00:11:03.270
and what's the largest
value that y can take?
00:11:03.270 --> 00:11:06.680
So what's the bottom boundary
and what's the top boundary?
00:11:06.680 --> 00:11:08.410
But if you look at
this region-- right?--
00:11:08.410 --> 00:11:10.290
life is a little
complicated here.
00:11:10.290 --> 00:11:12.604
Because if you're in the
left half of this region--
00:11:12.604 --> 00:11:13.770
what do I mean by left half?
00:11:13.770 --> 00:11:17.285
I mean if you're to the left
of this point of intersection--
00:11:17.285 --> 00:11:19.450
if you're at the left
of this line x equals
00:11:19.450 --> 00:11:22.430
square root of 2--
when you're over there,
00:11:22.430 --> 00:11:26.660
y is going from 0 to x.
00:11:26.660 --> 00:11:31.610
But if you're over in the
right part of this region,
00:11:31.610 --> 00:11:33.120
there's a different
upper boundary.
00:11:33.120 --> 00:11:34.250
Right?
00:11:34.250 --> 00:11:36.300
It's a different curve
that it came from.
00:11:36.300 --> 00:11:38.200
It has a different equation.
00:11:38.200 --> 00:11:44.410
So over here, y is going from
the x-axis up to the circle.
00:11:44.410 --> 00:11:47.860
So this is complicated, and what
does this complication mean?
00:11:47.860 --> 00:11:49.840
Well, it means that it's
not easy to write this
00:11:49.840 --> 00:11:51.770
as a single iterated integral.
00:11:51.770 --> 00:11:53.280
If you want to do
this in this way,
00:11:53.280 --> 00:11:55.520
you have to break the
region into two pieces,
00:11:55.520 --> 00:11:58.600
and write this double
integral as a sum of two
00:11:58.600 --> 00:12:00.090
iterated integrals.
00:12:00.090 --> 00:12:00.590
OK?
00:12:00.590 --> 00:12:03.930
So one iterated integral will
take care of the left part
00:12:03.930 --> 00:12:07.020
and one will take care
of the right part.
00:12:07.020 --> 00:12:08.665
So let's do the left part first.
00:12:11.550 --> 00:12:18.550
So here we're going to have a
iterated integral integrating
00:12:18.550 --> 00:12:20.120
with respect to y first.
00:12:20.120 --> 00:12:25.290
So to fixed value of x, we want
to know what the bounds on y
00:12:25.290 --> 00:12:25.834
are.
00:12:25.834 --> 00:12:27.500
And well, we can see
from this picture--
00:12:27.500 --> 00:12:30.180
when you're in this
triangle-- that y
00:12:30.180 --> 00:12:33.220
is going from the x-axis
up to the line y equals x.
00:12:33.220 --> 00:12:35.910
So that means the smallest
value that y can take is 0,
00:12:35.910 --> 00:12:38.225
and the largest value
that y can take is x.
00:12:38.225 --> 00:12:41.030
So here it's from 0 to x.
00:12:41.030 --> 00:12:43.050
And when you're
in this triangle,
00:12:43.050 --> 00:12:46.210
we need to know what the
bounds on x are, then.
00:12:46.210 --> 00:12:48.220
We need to know
the outer bounds.
00:12:48.220 --> 00:12:50.925
So we need to know the absolute
largest and smallest values
00:12:50.925 --> 00:12:51.967
that x can take.
00:12:51.967 --> 00:12:53.050
Well, what does that mean?
00:12:53.050 --> 00:12:55.791
We need to know the absolute
leftmost and absolute rightmost
00:12:55.791 --> 00:12:56.290
points.
00:12:56.290 --> 00:12:58.930
So the absolute leftmost
point is the origin.
00:12:58.930 --> 00:13:01.950
The absolute rightmost
is this vertical line
00:13:01.950 --> 00:13:03.540
x equals square root of 2.
00:13:03.540 --> 00:13:07.885
So over here, the
value of x is 0.
00:13:07.885 --> 00:13:11.570
And at the rightmost boundary
of this triangle, the value of x
00:13:11.570 --> 00:13:13.540
is the square root of 2.
00:13:13.540 --> 00:13:14.040
OK.
00:13:14.040 --> 00:13:16.090
So that's going to give
us the double integral
00:13:16.090 --> 00:13:20.130
just over this triangular
part of the region.
00:13:20.130 --> 00:13:22.420
Yeah?
00:13:22.420 --> 00:13:26.160
So now, we need to
add to this-- but I'm
00:13:26.160 --> 00:13:28.473
going to put it down on
this next line-- we need
00:13:28.473 --> 00:13:30.310
to add to this the
part of the integral
00:13:30.310 --> 00:13:34.580
over this little segment
of the circle here.
00:13:34.580 --> 00:13:38.750
The remainder of the region
that's not in that triangle.
00:13:38.750 --> 00:13:44.740
So for that, again, we're going
to write down two integrals,
00:13:44.740 --> 00:13:49.360
and it's going to be dy dx.
00:13:49.360 --> 00:13:50.052
Whew.
00:13:50.052 --> 00:13:51.315
We're nearly done, right?
00:13:54.560 --> 00:13:59.140
So y is inside, so we need
to know what the bounds on y
00:13:59.140 --> 00:14:01.041
are for a given value of x.
00:14:01.041 --> 00:14:02.540
So we need to know
for a given value
00:14:02.540 --> 00:14:06.820
of x, what are the bottom
and the topmost points
00:14:06.820 --> 00:14:08.490
of this region?
00:14:08.490 --> 00:14:10.450
So for a given value
of x, that means
00:14:10.450 --> 00:14:14.510
that y is going here between the
x-axis and between this circle.
00:14:14.510 --> 00:14:19.290
So the x-axis is y equals 0,
so that's the lower bound.
00:14:19.290 --> 00:14:22.650
So for the upper bound, we
need to know this circle.
00:14:22.650 --> 00:14:24.090
What is y on this circle?
00:14:24.090 --> 00:14:25.465
Well, the equation
of this circle
00:14:25.465 --> 00:14:27.080
is x squared plus
y squared equals 4,
00:14:27.080 --> 00:14:31.080
so y is equal to the square
root of the quantity 4 minus x
00:14:31.080 --> 00:14:31.580
squared.
00:14:36.440 --> 00:14:38.640
Where again, here we take
the positive square root,
00:14:38.640 --> 00:14:42.360
because this is a part of the
circle where y is positive.
00:14:42.360 --> 00:14:43.370
Yeah.
00:14:43.370 --> 00:14:46.490
If we were somehow on the
bottom part of the circle,
00:14:46.490 --> 00:14:49.222
then we would have to take a
negative square root there,
00:14:49.222 --> 00:14:51.180
but because we're on the
top part of the circle
00:14:51.180 --> 00:14:54.300
where y is positive, we
take a positive square root.
00:14:54.300 --> 00:14:54.980
OK, good.
00:14:54.980 --> 00:14:56.860
So those are the
bounds on y, and now we
00:14:56.860 --> 00:14:59.161
need to know the
absolute bounds on x.
00:14:59.161 --> 00:14:59.660
Yeah?
00:14:59.660 --> 00:15:01.780
So those are the bounds
on y in terms of x.
00:15:01.780 --> 00:15:03.870
And now because x
is the outer thing
00:15:03.870 --> 00:15:05.535
we're integrating
with respect to,
00:15:05.535 --> 00:15:07.530
we need the absolute
bounds on x.
00:15:07.530 --> 00:15:12.290
And you can see
in this circular--
00:15:12.290 --> 00:15:16.480
I don't really know what the
name for a shape like that is--
00:15:16.480 --> 00:15:20.020
but whatever that thing is, we
need to know what its leftmost
00:15:20.020 --> 00:15:21.750
and rightmost points are.
00:15:21.750 --> 00:15:25.070
We need to know the smallest and
largest values that x can take.
00:15:25.070 --> 00:15:30.620
And so its leftmost edge is this
line x equals square root of 2.
00:15:30.620 --> 00:15:33.660
And its rightmost edge is that
rightmost point on the circle
00:15:33.660 --> 00:15:35.490
there-- where the
circle hit the x-axis--
00:15:35.490 --> 00:15:37.390
and that's the value
when x equals 2.
00:15:40.710 --> 00:15:42.150
OK, so there you go.
00:15:42.150 --> 00:15:47.180
There's this last integral
written in the dy dx order,
00:15:47.180 --> 00:15:50.000
but we can't write it as a
single iterated integral.
00:15:50.000 --> 00:15:52.890
We need to write it as a sum of
two iterated integrals because
00:15:52.890 --> 00:15:54.930
of the shape of this region.
00:15:54.930 --> 00:15:55.430
All right.
00:15:58.600 --> 00:16:01.820
Let me just make one
quick, summary comment.
00:16:01.820 --> 00:16:05.570
Which is that if you're
doing this, one thing that
00:16:05.570 --> 00:16:09.167
should always be true, is
that these integrals, when
00:16:09.167 --> 00:16:10.750
you evaluate them--
so here, I haven't
00:16:10.750 --> 00:16:12.520
been writing an integrand.
00:16:12.520 --> 00:16:14.770
I guess the integrand has
always been 1, or something.
00:16:14.770 --> 00:16:19.000
But for any integrand,
the nature of this process
00:16:19.000 --> 00:16:26.050
is that it shouldn't matter
which order you integrate.
00:16:26.050 --> 00:16:28.810
You should get the same answer
if you integrate dx dy or dy
00:16:28.810 --> 00:16:29.680
dx.
00:16:29.680 --> 00:16:32.930
So one very low-level
check that you
00:16:32.930 --> 00:16:35.660
can make-- that you haven't done
anything horribly, egregiously
00:16:35.660 --> 00:16:38.060
wrong when changing the
bounds of integration--
00:16:38.060 --> 00:16:43.090
is that you can check that
actually these things evaluate
00:16:43.090 --> 00:16:44.010
the same.
00:16:44.010 --> 00:16:44.930
Yeah?
00:16:44.930 --> 00:16:46.910
Where you can
choose any function
00:16:46.910 --> 00:16:48.780
that you happen to
want to put in there--
00:16:48.780 --> 00:16:50.884
function of x and y-- and
evaluate this integral,
00:16:50.884 --> 00:16:53.550
and choose any function that you
happen to want to put in there,
00:16:53.550 --> 00:16:54.840
and evaluate those integrals.
00:16:54.840 --> 00:16:57.440
And see that you actually get
the same thing on both sides.
00:16:57.440 --> 00:17:00.360
Now one simple example
is that you could just
00:17:00.360 --> 00:17:02.180
evaluate the
integral as written,
00:17:02.180 --> 00:17:05.100
with a 1 written in there.
00:17:05.100 --> 00:17:07.400
And so in both cases,
what you should get
00:17:07.400 --> 00:17:09.320
is the area of the
region when you
00:17:09.320 --> 00:17:10.839
evaluate an integral like that.
00:17:10.839 --> 00:17:13.380
But you can also check with any
other function if you wanted.
00:17:19.910 --> 00:17:22.410
It won't show that what
you've done is right,
00:17:22.410 --> 00:17:25.980
but it will show if you've
done something wrong.
00:17:25.980 --> 00:17:29.517
That method will sometimes
pick it out, right?
00:17:29.517 --> 00:17:31.100
Because you'll
actually be integrating
00:17:31.100 --> 00:17:32.800
over two different regions,
and there's no reason
00:17:32.800 --> 00:17:34.290
you should get the same answer.
00:17:34.290 --> 00:17:36.680
So if you were to
compute these integrals
00:17:36.680 --> 00:17:38.140
and get different
numbers, then you
00:17:38.140 --> 00:17:40.440
would know that something
had gone wrong at some point
00:17:40.440 --> 00:17:43.780
for sure, and you'd have to go
and figure out where it was.
00:17:43.780 --> 00:17:45.633
I think I'll end there.