Q: Show that is equal to for every circle centered at .
First, I wrote
Here is where I am stuck. Firstly, I am assuming I will need to use the parameterization , , with .
So, I want to integrate along . I am not sure how to use the parameters and integrate along this curve.
Any help would be appreciated.
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?
Originally Posted by Ackbeet
Even so, we have not covered that material and his hint was to use greens thereom. We have just been doing integrals by seperating the real and imagenary parts into two seperate integrals, the one with the imaginary integrand has a constant i out front.
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.
Yep, that's exactly right. You're welcome, and have a good one!