Volume of a Sphere by Integrals

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

volume of a sphere

Figure 1. volume of a sphere generated by the rotation of a semi circle around x axis
\text{Volume} = \int_{x_1}^{x_2} \pi f(x)^2 dx \\
Substitute f(x) by its expression √(R 2 - x 2).
= \int_{-R}^{R} \pi (\sqrt{R^2 - x^2})^2 dx \\
Simplify.
= \int_{-R}^{R} \pi (R^2 - x^2) dx \\
Integrate.
= \pi\left [R^2 x - x^3/3 \right ]_{-R}^R \\
Evaluate integral.
= \pi\left [ (R^3 - R^3/3) - (-R^3 + R^3/3) \right ] = \dfrac{4}{3} \pi R^3
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.


More Links and References

integrals and their applications in calculus.
Area under a curve.
Area between two curves.
Find The Volume of a Solid of Revolution.
Volume by Cylindrical Shells Method.