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

Problem

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

Solution

The graph of \( y = \sqrt{R^2 - x^2} \) from \( x = -R \) to \( x = R \) is shown below. Let \( f(x) = \sqrt{R^2 - x^2} \), 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 \( \sqrt{R^2 - x^2} \).
\[
= \int_{-R}^{R} \pi \left(\sqrt{R^2 - x^2}\right)^2 dx \\
\]
Simplify.
\[
= \int_{-R}^{R} \pi (R^2 - x^2) dx \\
\]
Integrate.
\[
= \pi\left [R^2 x - \frac{x^3}{3} \right ]_{-R}^R \\
\]
Evaluate integral.
\[
= \pi\left [ \left(R^3 - \frac{R^3}{3}\right) - \left(-R^3 + \frac{R^3}{3}\right) \right ] = \frac{4}{3} \pi R^3
\]
This is the well-known formula for the volume of a sphere. If you revolve a semicircle of radius \( R \) around the x-axis, it will generate a sphere of radius \( R \).