Question

Evaluate the integral Find the volume of material remaining in a hemisphere of radius 2 after a cylindrical hole ofradius 1 is drilled through the center of the hemisphere perpendicular to its base.

Answer 1

You want to find the volume inside the hemisphere
`x^2+y^2+z^2=4` (i.e. inside the sphere but above the plane
`z=0`) and outside the cylinder
`x^2+y^2=1`. Call this region
`R`.

In cylindrical coordinates, we have

`\displaystyle\iiint_R\mathrm dV=\int_0^(2\pi)\int_1^2\int_0^(√(4-r^2))r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta`

`\displaystyle=2\pi\int_1^2 r√(4-r^2)\,\mathrm dr`

`\displaystyle=-\pi\int_3^0\sqrt u\,\mathrm du`

(where
`u=4-r^2`)

`\displaystyle=\pi\int_0^3\sqrt u\,\mathrm du`

`=\frac{2\pi}3u^(3/2)\bigg|_0^3=2\sqrt3\,\pi`