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

$\displaystyle\iiint_R\mathrm dV=\int_0^{2\pi}\int_1^2\int_0^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$

$\displaystyle=2\pi\int_1^2 r\sqrt{4-r^2}\,\mathrm dr$

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

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

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

$=\dfrac{2\pi}3u^{3/2}\bigg|_0^3=2\sqrt3\,\pi$