I am learning multiple integrals (Double and Triple Integral) and need help understanding a solution given in the book.
In first question, it is asked to find the area lying inside the circle $r=a\sin\theta$ and outside the cardioid $r=a(1-\cos\theta)$. using double integrals. In this question how do i find the range for $r$ for integration ?
You don't need the limits over $r$; rather, you need them over $\theta$. The circle lies outside the cardioid when $\theta \in [0,\pi/2]$; you can see this from a plot or by simply looking at the equations themselves. The area you seek is then
$$\int_0^{\pi/2} d\theta \, \int_{a(1-\cos{\theta})}^{a \sin{\theta}} dr \, r$$
I get as a result
$$\left ( 1-\frac{\pi}{4}\right ) a^2$$