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`

`=\frac{218\pi}3`