Question

Find volume of cone z=sqrt(x^2+y^2) that is bounded with z=5 and z=7

Answer 1

Convert to cylindrical coordinates first, using

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

Then the volume is

$\displaystyle\int_{\zeta=5}^{\zeta=7}\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\zeta}r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta$
$=\displaystyle\pi\int_{\zeta=5}^{\zeta=7}\zeta^2\,\mathrm d\zeta$
$=\dfrac{218\pi}3$