# 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 $$y = \sqrt{R^2 - x^2}$$ 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 $$R$$.

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 .