Volume of a Spherical Cap

The volume of a spherical cap is found using integrals and the method of disks used in "volume of a Solid of revolution".
A spherical cap is defined as as a portion of the sphere cut by a plane. spherical cap as portion of a sphere


We now consider a spherical cap from a sphere of radius R and height \( h \). A spherical cap may be generated by revolving the curve of \( y = \sqrt{R^2 - x^2} \), which is half a circle, around the x-axis with \( x \) in the range \( R-h \le x \le R \).

spherical cap in 2D

Consider the small disk (in broken lines) having a width \( dx \) and a radius equal to \( y \). The volume of the disk is given by \( \pi y^2 dx \) and thefore the integral over \( x \) in the ramge \( [R-h, R ] \) gives the volume of the cap.
            \( \displaystyle \text{Volume} = \int_{R-h}^{R} \pi y^2 dx \)
Since \( y = \sqrt{R^2 - x^2} \), the volume is given by
            \( \displaystyle \text{Volume} = \pi \int_{R-h}^{R} (R^2 - x^2) dx \)
Evaluate the integral
            \( \displaystyle \text{Volume} = \pi \left[R^2 x - \dfrac{1}{3}x^3\right]_{R-h}^{R} \)
Evaluate the above expression on the right
            \( \displaystyle \text{Volume} = \pi \left\{ \left(R^3 - \dfrac{1}{3}R^3\right) - \left(R^2 (R-h) - \dfrac{1}{3}(R-h)^3\right) \right\} \)
Simplify the above to obtain the volume
            \[ \Large \displaystyle \color{red} {\text{Volume} = \dfrac{\pi}{3}( 3 Rh^2-h^3) }\]

More Links and References

Volume of a Solid of Revolution
Area under a curve .
Area between two curves .
integrals and their applications in calculus.