Problem
Express the double integral $\iint_S f(x,y) \,dx \,dy$ as an iterated integral in polar coordinates, where $S=\{(x,y)|x^2\leq y,-1 \leq x\leq 1\}$.
Doubt
I have difficulty finding range of $r$ and $\theta$.
$r^2 \cos^2(\theta) \leq r \sin(\theta) \leq 1$
$-1 \leq r \sin(\theta) \leq 1$
One may divide the area to three parts: \begin{cases} 0\leq\theta\leq\dfrac{\pi}{4};\ 0\leq r\leq\sin\theta\sec^2\theta,\\ \dfrac{\pi}{4}\leq\theta\leq\dfrac{3\pi}{4};\ 0\leq r\leq\sec\theta,\\ \dfrac{3\pi}{4}\leq\theta\leq\pi;\ 0\leq r\leq\sin\theta\sec^2\theta. \end{cases} then \begin{align} \int_{-1}^{1}\int_{x^2}^{1} f(x,y)\ dy\ dx &= \int_{0}^{\pi/4}\int_{0}^{\sin\theta\sec^2\theta} f(r\cos\theta, r\sin\theta)\ r\ dr\ d\theta \\ &+ \int_{\pi/4}^{3\pi/4}\int_{0}^{\sec\theta} f(r\cos\theta, r\sin\theta)\ r\ dr\ d\theta \\ &+ \int_{3\pi/4}^{\pi}\int_{0}^{\sin\theta\sec^2\theta} f(r\cos\theta, r\sin\theta)\ r\ dr\ d\theta \end{align}