# Find the volume of the Region (Triple Integral with Cylindrical Coordinates)

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.

• why is there $r^3$? shouldnt there just be an r in the change of coordinates , im assuming your integrating with the caratheristic function. – Something Apr 16 '19 at 22:46

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 $$486$$.