# 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 \\

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 .