How to describe this region in polar coordinates?

Question

$D=\{(x,y)\in\mathbb{R}^2: x^2+y^2\leq 9 \text{ and } y\in [-3,1]\}$ I know how the region looks like but when $\theta \in [\sin^{-1}(\frac{1}{3}),\pi -\sin^{-1}(\frac{1}{3})]$ I don't know how to describe how the radius $r$ moves.

Answer 1

If you were to integrate over this region, you would split it into two integrals: one integrating on the circle minus a sector, and then on the rest. For the latter case, the bound for $r$ depends on $\theta$. $y = r\sin(\theta)$, so we have $$0 \leq y \leq 1 \iff 0\leq r\sin(\theta)\leq 1 \iff 0 \leq r \leq \csc(\theta)$$ And so your integral (if you're concerned with integration here) would look like $$\int_{\sin^{-1}(\frac{1}{3})}^{\pi-\sin^{-1}(\frac{1}{3})} \int_{0}^{\csc(\theta)}f(r, \theta) dr\space d\theta$$ In other words, the region can be described as {$(r, \theta) | 0\leq r\leq 3, b\leq \theta \leq a+2\pi$}$\cup${$(r, \theta) | 0\leq r \leq \csc(\theta), a\leq \theta \leq b$}, where $a$ and $b$ are the appropriate bounds for $\theta$.