Find the volume of a sphere using integrals and the disk method.

Problem

Find the volume of a sphere generated by revolving the semicircle y = √
(R^{ 2} - x^{ 2}) around the x axis.

Solution

The graph of y = √(R^{ 2} - x^{ 2}) from x = - R to x = R is shown below. Let f(x) = √(R^{ 2} - x^{ 2}), the volume is given by formula 1 in Volume of a Solid of Revolution

\text{Volume} = \int_{x_1}^{x_2} \pi f(x)^2 dx \\

Substitute f(x) by its expression √(R^{ 2} - x^{ 2}).

This is the very well known formula for the volume of the sphere. If you revolve a semi circle of radius R around the x axis, it will generate a sphere of radius R.