Question

Question part points submissions used use cylindrical coordinates. evaluate x2 + y2 dv, e where e is the region that lies inside the cylinder x2 + y2 = 4 and between the planes z = 3 and z = 7.

Answer 1

$\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=2}\int_{z=3}^{z=7}r^3\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=8\pi\displaystyle\int_0^2r^3\,\mathrm dr=2\pi(2^3-0^3)=16\pi$