Question

Find the volume of the wedge-shaped region (Figure 1) contained in the cylinder x2+y2=81 and bounded above by the plane z=x and below by the xy-plane.

Answer 1

In cylindrical coordinates, the volume is given with


`\displaystyle\int_(\theta=-\pi/2)^(\theta=\pi/2)\int_(r=0)^(r=9)\int_(z=0)^(z=r\cos\theta)r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta`

`=\displaystyle\int_(\theta=-\pi/2)^(\theta=\pi/2)\int_(r=0)^(r=9)r^2\cos\theta\,\mathrm dr\,\mathrm d\theta`

`=\displaystyle\left(\int_(\theta=-\pi/2)^(\theta=\pi/2)\cos\theta\,\mathrm d\theta\right)\left(\int_(r=0)^(r=9)r^2\,\mathrm dr\right)`

`=486`