Find the volume of a sphere using integrals and the disk method.
In what follows, \( C \) represents the constant of integration.
Problem & Derivation
Problem: Find the volume of a sphere generated by revolving the semicircle \( y = \sqrt{R^2 - x^2} \) around the x-axis.
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:
\[ \large{\text{Volume} = \color{red}{\int_{x_1}^{x_2} \pi (f(x))^2 \, dx}} \]
Setting up the definite integral with bounds from \( -R \) to \( R \):
\[ \text{Volume} = \int_{-R}^{R} \pi f(x)^2 \, dx \]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 the 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. Revolving a semicircle of radius \( R \) around the x-axis generates a sphere of radius \( R \).