Math Help - Changing the limits of integration for this u-sub...

1. Choosing limits for the area of a lemniscate

Hi all,

I started with the equation:

What I'm doing is calculating the area enclosed by this lemniscate.

What I determined is that the positive cycle I wanted to look at was at 3pi/4 to 5pi/4.

The problem is that my integral is ending up at 0. I suspect it's because you can't have a negative answer from a square root in the reals. I thought I could leave it alone since the section I was looking at was positive:

What I have to do for this problem is a u-substitution. When I do that, that changes it to cos(u) which then falls into an area where I'd have to split the integral up to properly evaluate it. What I'm wondering is if I can change the limits of integration to compensate for that somehow, or if not do I just have to look and see what cos(u) is doing between those limits and rewrite the integral accordingly?

Thanks in advance for your help!

2. Hmmm...afterthought....maybe thats a poor choice for my limits of integration. I'm going to rename this thread since that's what it's really about I think.

3. Update...

I took the area for one of the quadrants and multiplied it by 4...got 1 for an answer, which agrees with what I've read that the area will be a^2.

Hopefully that's right.