Evaluate the double integral integral ∫∫ y dxdy. The plane region is (x-1)^2+(y-1)^2 <=8.

Answer 1

The integration region is a circle centered at (1,1) of radius r = 2root(2).

Change to polar coordinates adapted to that i.e. centered at (1,1):

x = 1 + r cosθ

y = 1 + r sinθ

The area element:

dA = rdθdr

This can be proven by evaluating the Jacobian of that change of variables. It goes exactly like the standard polar coordinates centered at (0,0).

The integral in the new coordinates becomes:

∫∫(1 + r sinθ) dθrdr

Integrate first over θ from 0 to 2π, and then over r from 0 to 2root(2). The integral is very easy to take in those polar coordinates.