00:03
Okay, for this question, they want us to take the double integral over the region of the surface 3xy d8, and they describe the surface to us in english.
00:15
It's going to be bound by the curves y is equal to 2 minus x.
00:20
It's going to be bound by the horizontal line y is equal to 0, and it's going to be bound by the curve, x is equal to 4 minus y squared.
00:29
Okay, they want us to depict, oh, wait, quadrant 1, that's the restriction.
00:33
And now they want us to depict the region that we're integrating over.
00:37
And let's see.
00:39
The x -axis is one of the bounds.
00:40
And then we shift up two units.
00:42
And we look down one step at a time.
00:47
And then we have a parabola with respect to y.
00:50
Let's see, four units.
00:52
And it goes like this.
00:55
And like this below the x -axis.
00:58
So it looks like the region we care about is this one.
01:02
So we have curves on the right and left.
01:04
So let's see our final integration.
01:05
Will be with respect to y.
01:07
This intersection point is probably 2.
01:10
If we set 0 to both, let's see.
01:12
This curve is 0, the y -intercept.
01:15
And this curve will have y -squared is equal to 4, which is plus or minus 2.
01:20
Okay, the only one we care about is plus 2 since it's above the x -axis.
01:24
So we are integrating from 0 to 2 with respect to y.
01:27
And our right and left bounds, let's see.
01:30
So the right is 4 minus y squared.
01:32
And now i need to write our other curve in terms of x.
01:36
In terms of y.
01:37
Okay.
01:38
And now we're going to integrate that surface, dx, d, d, y.
01:42
So let's see.
01:43
We'll compute the inner integral.
01:48
3 over 2, x squared y, evaluated at 4 minus y squared and 2 minus y.
01:55
Okay, so we're going to have to do some, let's see, of the x...