I don't think your general method is correct. I would parametrize the circle like this:
Let for constant and Thus, What does your integral become?
Q: Show that is equal to for every circle centered at .
First, I wrote
Now,
.
So,
.
Thus,
Here is where I am stuck. Firstly, I am assuming I will need to use the parameterization , , with .
So, I want to integrate along x-x_{0})^{2}+(y-y_{0})^{2}=r\}" alt="C=\{(x,y)x-x_{0})^{2}+(y-y_{0})^{2}=r\}" />. I am not sure how to use the parameters and integrate along this curve.
Any help would be appreciated.
Thanks
You can do Green's Theorem. So, in reviewing your work, I'd say you're good up to here:
Now, what you want to do is apply Green's Theorem on each integral. Green's theorem is going to reduce your integrals quite a bit. Recall that Green's Theorem states that
Once you have constructed your double integrals, I'd recommend converting to polar coordinates. The result pops out fairly readily.
Ahhhh, I see. The first integral will go to zero and I will end up with the integrand 2 after taking the partails of the other chunk and applying greens thereom. So, if convert to polar and integrate over the region I end up with I am looking for after distributing the and the .
Thanks for the help, I appreciate it. I also carried out the method you mentioned in your first post and got the answer that way.
Thanks again.