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 a portion of a sphere cut off by a plane.
Derivation of the Spherical Cap Volume Formula
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 represents half a circle, around the x-axis with \( x \) in the range \( R - h \le x \le R \).
Consider a small disk (shown with 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 therefore the integral over \( x \) in the range \( [R - h, R] \) gives the total volume of the cap:
\[ \text{Volume} = \int_{R-h}^{R} \pi y^2 \, dx \]Since \( y = \sqrt{R^2 - x^2} \), the volume integral becomes:
\[ \text{Volume} = \pi \int_{R-h}^{R} (R^2 - x^2) \, dx \]Evaluating the integral:
\[ \text{Volume} = \pi \left[ R^2 x - \dfrac{1}{3}x^3 \right]_{R-h}^{R} \]Substitute the upper and lower limits:
\[ \text{Volume} = \pi \left\{ \left( R^3 - \dfrac{1}{3}R^3 \right) - \left( R^2(R - h) - \dfrac{1}{3}(R - h)^3 \right) \right\} \]Simplifying the above algebraic expression yields the final formula for the volume:
\[ \Large \displaystyle \color{red}{\text{Volume} = \dfrac{\pi}{3}(3Rh^2 - h^3)} \]