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.
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 x2 + z2 = r2. This means so the area of the circle is
and the volume of the slice is
π(r2 – x2) Δ x.
The volume of the sphere is
We'll evaluate the second integral. Drats. Karma finally caught up.