Question

find the volume of the solid bounded below by the xy plane, on the sides by the sphere rho=2 and above by the cone fee= pie over 3

Answer 1

`\displaystyle\int_(\theta=0)^(\theta=2\pi)\int_(\varphi=\pi/3)^(\varphi=\pi/2)\int_(\rho=0)^(\rho=2)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta`

`=\displaystyle\left(\int_(\theta=0)^(\theta=2\pi)\mathrm d\theta\right)\left(\int_(\varphi=\pi/3)^(\varphi=\pi/2)\sin\varphi\,\mathrm d\varphi\right)\left(\int_(\rho=0)^(\rho=2)\rho^2\,\mathrm d\rho\right)`

`=\displaystyle2\pi\left(-\cos\varphi\bigg|_(\varphi=\pi/3)^(\varphi=\pi/2)\right)\left(\frac13\rho^3\bigg|_(\rho=0)^(\rho=2)\right)`

`=\frac{8\pi}3`