00:01
So we're going to be evaluating this integral here.
00:04
And the first thing we want to do is acknowledge what the f of x, y, g of x, y is.
00:13
And it's clear to us that that's going to be a z squared.
00:19
So right here we'll put z squared.
00:30
And then we know that on the inside we're given that since x equals y squared plus, z squared the way we can write this will be partial derivative with respect to y and the partial derivative with respect to z so this will ultimately become once we square all this we'll get a 4 y squared plus 4 z squared and ultimately the reason why that's the case is because if we go back to this um right here what we did instead was we made this in terms of y and z, which means we change these partial derivatives to be y and z.
01:30
So as such, we'll have z squared, like we said, and then this will be 4 y squared plus 4 z squared, and this will be d y d z.
01:50
Then we know that the region inside the circle is the region is inside the circle y squared plus z squared is less than equal to one which makes it easier to use polar coordinates we're just going to essentially treat y as if it's x and z as if it's y so because of that what we'll have is z squared will now be r squared sine squared theta and then 4 y squared plus 4 z squared plus 1 will just be 1 plus for r squared and then outside of this we'll have r, the r, d -theta...