Finding the area of one loop of the lemniscate

1. Nov 20, 2012

annpaulveal

1. The problem statement, all variables and given/known data

Find the area of the region bounded by one loop of the lemniscate r2 = a2sin(2θ) with a > 0 using double integration.

2. Relevant equations

3. The attempt at a solution

I was able to figure out the limits of integration for theta (0 to ∏/2), but what would my limits be for r?

2. Nov 20, 2012

haruspex

The limits for r will be functions of theta. Integrate wrt r first. When you plug in the limits, those functions will come into the integrand. Then you can integrate wrt theta.

3. Nov 20, 2012

annpaulveal

I'm sorry, I really don't understand. How can I integrate wrt r without setting the limits of my integrand first? What would those limits be?

4. Nov 20, 2012

LCKurtz

When you integrate in the r direction, r goes from r = 0 to the r on the curve, which is a function of $\theta$.

5. Nov 20, 2012

haruspex

For a given value of theta, what is the smallest value of r within the region? What is the largest value with the region?