Chapter 6.2, Problem 31E

To determine

To calculate: The volume of the right circular cone.

Answer to Problem 31E

The volume of the circular cone is $\underset{\_}{\frac{1}{3}\pi r^{2}h}$.

Explanation of Solution

Given information:

The right circular cone with height h and base radius r.

The dimensions of the right circular cone as shown in Figure 1.

Refer to Figure 1.

Consider that the right circular cone is obtained by rotating the line $y=\frac{r}{h}x$ about x-axis.

The region lies between $a=0$ and $b=h$.

Calculation:

The expression to find the volume of the right circular cone as shown below.

$V={\displaystyle {\int }_a^{b}A\left(x\right)dx}$ (1)

Find the area of the right circular cone as shown below.

$\begin{array}{c}A\left(x\right)=\pi r^2\\ =\pi {\left(\frac{r}{h}x\right)}^2\\ =\frac{\pi r^2{x}^2}{h^2}\end{array}$

Substitute 0 for a, h for b, and $\frac{\pi r^2{x}^2}{h^2}$ for $A\left(x\right)$ in Equation (1).

$\begin{array}{c}V={\displaystyle {\int }_0^h\frac{\pi r^2{x}^2}{h^2}dx}\\ =\frac{\pi r^2}{h^2}{\left(\frac{x^3}{3}\right)}_0^h\\ =\frac{\pi r^2}{h^2}\left(\frac{h^3}{3}-0\right)\\ =\frac{1}{3}\pi r^{2}h\end{array}$

Therefore, the volume of the right circular cone is $\underset{\_}{\frac{1}{3}\pi r^{2}h}$.