Question

Evaluate x2 dv, e where e is the solid that lies within the cylinder x2 + y2 = 4, above the plane z = 0, and below the cone z2 = 9x2 + 9y2.

Answer 1

Convert to cylindrical coordinates, setting

$\begin{cases}x=r\cos t\\y=r\sin t\\z=z\end{cases]$

Then

$\displaystyle\iiint_Ex^2\,\mathrm dV=\int_{t=0}^{t=2\pi}\int_{r=0}^{r=2}\int_{z=0}^{z=3r}r^3\cos^2t\,\mathrm dz\,\mathrm dr\,\mathrm dt=\frac{96\pi}5$