Question

Evaluate ,where e is the solid that lies between the cylinders x^2+y^2=1 and x^2+y^2=16, above the xy-plane, and below the plane z=y+4

Answer 1

Assuming you're just computing the volume, i.e.

$\displaystyle\iiint_E\mathrm dV$

we should convert to cylindrical coordinates first, setting

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

so that the volume is given by

$\displaystyle\iiint_E\mathrm dx\,\mathrm dy\,\mathrm dz=\int_{\theta=0}^{\theta=2\pi}\int_{r=1}^{r=4}\int_{\zeta=0}^{\zeta=r\sin\theta+4}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta$

which has a value of $60\pi$.