Chapter 8.3, Problem 71E

To determine

To derive : the formula of volume of a hemisphere.

Explanation of Solution

Given information : The volume of a hemisphere of radius $R$ by comparing its cross section with the cross sections of a solid right circular cylinder of radius $R$ and height $R$ from which a solid right circular cone of base radius $R$ and height $R$

has been removed as suggested by the figure:

  

The hemisphere cross sectional area is given by:

  $\pi {\left(\sqrt{R^2-h^2}\right)}^2$

The cross section of the right circular cylinder with cone removed is $\pi R^2-\pi h^2$

Since these area are equal, the volumes of the solid formed are equal by cavalier’s theorem. Therefore, the volume of the hemisphere is given by the volume of the cylinder less the volume of the cone.

  $\begin{array}{l}\pi R^3-\frac{1}{3}\pi R^3\\ \Rightarrow \frac{2}{3}\pi R^3\end{array}$