A hole of diameter d is drilled through a sphere of radius r in such a way that the axis of the hole passes through the centre of the sphere. Find the volume of the solid that remains.

I drew a picture of a sphere centred on the cartesian plane, with a hole going through its centre (in the shape of a cylinder).

I already knew that the volume of the sphere was $\displaystyle \frac {4}{3}\pi r^{3}$. I then worked out the volume of the hole (cylinder):

$\displaystyle

\int_{-r}^{r} \frac{1}{4}\pi d^{2}dy$

$\displaystyle = \frac{1}{2}\pi d^{2} r

$

Volume of solid remaining = $\displaystyle \frac {4}{3}\pi r^{3} - \frac{1}{2}\pi d^{2} r $

but the correct answer was $\displaystyle \frac{1}{6}\pi (4r^{2} - d^{2})^{\frac{3}{2}}$

I cant see where I went wrong cause surely the objective of this problem is to minus the hole (cylinder) from a sphere to get the remaining volume?