Volume of a Sphere by Integrals

Volume of a Sphere (equation derived with calculus)
Volume of a Sphere (equation derived with calculus)

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
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 √(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.
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