Given a solid sphere of radius , what is the volume of material from a hole of radius through the center of the sphere.

First, the volume of a sphere of radius is given by

Then, the volume of a sphere with a hole drilled in it is the volume of the solid of revolution generated by the region between and from to . Denoting this volume by we then have,

Thus, the volume of the material removed from the sphere by drilling a hole in it is given by

*Related*

The third step of your simplification of V(T) has an extra π (Pi) coefficient in the second term.

Thanks! Fixed.