What is the area inside the curves $r = \cos\theta$ and $r = \sin\theta$?

Question

I have set up an integral's bounds in polar coordinates: $\int_{0}^{\pi/2}\int_{\sin\theta}^{\cos\theta}drd\theta$, but I do not know what function to integrate (what goes inside the integral symbols). I included an image of my thought process of how I approached this problem setup.

Answer 1

If you want the area between the two circles it must be that

$r\le \min(\cos \theta, \sin \theta)$

or, $r \le\begin{cases}\sin \theta& 0\le \theta\le \frac \pi4\\\cos \theta & \frac \pi4<\theta\le \frac \pi 2\end{cases}$

The region is symmetric across the line $\theta = \frac \pi4$ So, we can integrate from $0$ to $\frac \pi4$ and double the result.

$2\int_0^{\frac {\pi}{4}}\int_0^{\sin\theta} r\ dr\ d\theta\\ 2\int_0^{\frac {\pi}{4}} \frac 12 r^2 |_0^{\sin \theta}\ d\theta\\ \int_0^{\frac {\pi}{4}} \sin^2\theta\ d\theta$