Write an integral expression for the volume of a sphere with radius *r*. Evaluate your expression to get a formula for the volume of a sphere.

Answer

This is like the example but with no numbers. Yes, there's a theme here. We told you we would give you the power to derive any area formula you wanted. We will slice the sphere vertically and use *x* to denote the position of a slice. Each slice has thickness Δ *x*. If we cut the sphere down the middle we can see the radius of the slice is *z* where *x*^{2} + *z*^{2} = *r*^{2}. This means so the area of the circle is

and the volume of the slice is

π(*r*^{2} – *x*^{2}) Δ *x*.

The volume of the sphere is

or

We'll evaluate the second integral. Drats. Karma finally caught up.