Question

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

Answer 1

Perhaps the best way to compute this volume is by converting to cylindrical coordinates.

If the region is denoted
`D`, then the volume is given by


`V=\displaystyle\iiint_D\mathrm dV`

`V=\displaystyle\int_0^4\int_0^(2\pi)\int_(z/4)^zr\,\mathrm dr\,\mathrm d\theta\,\mathrm dz`

`V=\displaystyle\pi\int_0^4\left(r^2\bigg|_(r=z/4)^(r=z)\right)\,\mathrm dz`

`V=\displaystyle\pi\int_0^4\left(z^2-(z^2)/(16)\right)\,\mathrm dz`

`V=\displaystyle(15\pi)/(16)\int_0^4z^2\,\mathrm dz`

`V=\displaystyle(5\pi)/(16)z^3\bigg|_(z=0)^(z=4)`

`V=20\pi`