Finding the area of one loop of the lemniscate

[annpaulveal]

Homework Statement

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

Homework Equations

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?

 

[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.

 

[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?

 

[LCKurtz]

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?

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

 

[haruspex]

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?

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