Question

Evaluate the triple integral ∭z dv where e is the solid bounded by the cylinder y^2+z^2=144 and the planes x=0,y=3x and z=0 in the first octant

Answer 1

We can use a nonstandard set of cylindrical coordinates to set up an easy integral. Take

$\begin{cases}x=\xi\\y=r\cos\theta\\z=r\sin\theta\end{cases}$

and we get

$\displaystyle\iiint_Ez\,\mathrm dV=\int_{\theta=0}^{\theta=\pi/2}\int_{r=0}^{r=12}\int_{\xi=0}^{\xi=\frac13r\cos\theta}(r\sin\theta)r\,\mathrm d\xi\,\mathrm dr\,\mathrm d\theta$
$\displaystyle=\frac13\int_{\theta=0}^{\theta=\pi/2}\int_{r=0}^{r=12}r^3\sin\theta\cos\theta\,\mathrm dr\,\mathrm d\theta$
$\displaystyle=\frac16\left(\int_{\theta=0}^{\theta=\pi/2}\sin2\theta\,\mathrm d\theta\right)\left(\int_{r=0}^{r=12}r^3\,\mathrm dr\right)$
$=864$