Question

Calculate the volume of the solid, bounded by the surfaces: z=4x2 +4y2; z=x2+y2; z=4.

Answer 1

In cylindrical coordinates, the volume is given by the integral


`\displaystyle\int_(\theta=0)^(\theta=2\pi)\int_(z=0)^(z=4)\int_(r=√(z/4))^(r=\sqrt z)r\,\mathrm dr\,\mathrm dz\,\mathrm d\theta`

`=\displaystyle2\pi\int_(z=0)^(z=4)\int_(r=√(z/4))^(r=\sqrt z)r\,\mathrm dr\,\mathrm dz`

`image`

`=\displaystyle\pi\int_(z=0)^(z=4)\left(z-\frac z4\right)\,\mathrm dz`

`=\displaystyle\frac{3\pi}4\int_(z=0)^(z=4)z\,\mathrm dz`

`=\displaystyle\frac{3\pi}8z^2\bigg|_(z=0)^(z=4)`

`=6\pi`