Volume of a Sphere by Integrals
Find the volume of a sphere using integrals and the disk method.
ProblemFind the volume of a sphere generated by revolving the semicircle y = √ (R 2 - x 2) around the x axis.SolutionThe 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 Referencesintegrals 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. |
