Question

Where e is enclosed by the planes z = 0 and z = x + y + 9 and by the cylinders x2 + y2 = 4 and x2 + y2 = 16?

Answer 1

Best to use cylindrical coordinates. The volume (assuming that's what you're supposed to find) of the region is given by the triple integral

$\displaystyle\iiint_E\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=2}^{r=4}\int_{z=0}^{z=r(\cos\theta+\sin\theta)+9}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=2}^{r=4}(r^2(\cos\theta+\sin\theta)+9r)\,\mathrm dr\,\mathrm d\theta$
$=108\pi$