00:06
Our integral is x plus y sine x minus y da over region d bounded by lines y equals x, y equals x minus 8, y equals negative x, y equals negative x plus 6.
00:33
We're going to do a change of variables.
00:35
We're going to let u equal x plus y and then v equals x minus y.
00:51
So that means y equals x becomes v equals 0, y equals x minus 8 becomes v equals 8, y equals negative x is u equals 0, y equals negative x plus 6 is u equals 6.
01:07
Then express x and y in terms of u and v.
01:16
U plus v divided by 2 equals x, u minus v divided by 2 equals y.
01:26
Then we calculate the jacobian.
01:34
Jacobian is partial of x with respect to u, partial of x with respect to v, partial of y with respect to u, partial of y with respect to v.
01:53
So we're going to have one half, one half, one half, negative one half.
02:01
The determinant is negative one half.
02:06
Then we rewrite the integral.
02:11
Inside we should have u sine v that multiplies by the absolute value of the jacobian.
02:21
Du dv, u going from 0 to 6, v 0 to 8...