Question: Find the volume of the region that is contained by the cylinder $x^2+y^2=81$, bounded above by $z=x$ and below by the $xy$-plane.
I have tried the integral $\int_{0}^{2pi}\int_{0}^{9}\int_{0}^{rcos\theta}r^3dzdrd\theta$ and got $0$, same when $0\leq\theta\leq\pi$. Then I tried $\int_{0}^{pi/2}\int_{0}^{9}\int_{0}^{rcos\theta}r^3dzdrd\theta$ and got 11809.8. Which is also wrong.
If you are bounded below by the $xy$ plane and bounded above by the $z=x$ plane, then $z\geqslant0$ and $z\leqslant x$. So, $x\geqslant 0$ and so, since $x=\rho\cos\theta$, $\theta\in\left[-\frac\pi2,\frac\pi2\right]$. So, you consider the integral$$\int_{-\frac\pi2}^{\frac\pi2}\int_0^9\int_0^{\rho\cos\theta}\rho\,\mathrm dz\,\mathrm d\rho\,\mathrm d\theta.$$You should get $486$.