00:01
So we're going to be using this formula right here, which is f .d.
00:04
D .s.
00:08
Is equal to the double integral over the region of negative p, gx, minus q, gy, plus rda.
00:23
So first thing we'll do is calculate our g sub x and g sub y, given that g of x, y is equal to the square root of x squared plus y squared.
00:35
So when we evaluate it, we get our g sub x is x over the square root of x squared plus y squared, and our g sub y is the square root of, or is y over the square root of x squared plus y squared.
00:48
So when we multiply everything together, we end up getting the double integral over the region d of the square root of x squared plus y squared plus z cubed da.
01:05
And keep in mind da is dx, d -y, or d -y -d -x, so we can't have the z.
01:11
So instead, we're going to replace it with this because this is the same thing as z.
01:17
So when we do that, we'll put it in place.
01:24
And what we'll end up getting as a result when we substitute it is this double integral of x squared plus y squared plus 1 times the square root of x squared plus y squared d a then um we can convert this to polar coordinates um so when we do that we recognize that the radius is between the radius of the region is between one and three and um our theta values between zero and two theta um so what we'll do is we'll put that here 0 and 2 pi that is.
02:10
So we'll have r squared plus 1...