Question 66 for GR9768

[moo5003]

66. The line integral [x^2 dy -2y dx] over the circle x^2+y^2=9.

Now I tried using Greene's theorem on this but got bogged down when doing the double integral.

M = -2y
N = x^2

Integral of 2x + 2 dA.

First integrate with respect to y with limits +/- Root(9-x^2)

Then you get the integral from -3 to 3 of:

4(x+1)Root(9-x^2) <--- I stopped here.

Instead of using Greene’s theorem I started over parameterizing the equation in terms of t and then solving it directly using a few trig substitutions along the way (answer was 18pi). I was wondering what you guys think the best way to tackle this problem is?

[CoCoA]

A circle with center at zero is a very nice object in polar. Try converting the double integral of (2x+2)dA to polar.

[moo5003]

Thanks, completley forgot about polar. Made it much easier ^_^.

[ngoc]

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?

[ngoc]

Ah ok -- never mind. I figured out that I made a mistake in both methods (polar coord with Green's theorem, AND integration directly). So 18\pi it is.