Even with polar I still get $36\pi$ (twice the answer). I'm not sure why??
Also, I doubled check my answer by reasoning that:
\int_C (-y)dx = area of the circle = 9\pi.
Hence 2\int_C (-y)dx = 18\pi
and the left over is:
\int x^2 dy, (which is not a conservative vector field), and in fact can be calculated to be 18\pi. So the ansewr is 36\pi again. Please tell me where I went wrong?