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
formula for volume of a solid generated by revolving a triangle around  the x axes
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 \( \sqrt{R^2 - x^2} \). \[ = \int_{-R}^{R} \pi \left(\sqrt{R^2 - x^2}\right)^2 dx \\ \] 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 \( R \).


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 .
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