Q 66 - Calculus

Question

Use a definite integral with the shell method to prove that the volume formula $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ holds for a cone of radius $3$ and height $5$. 

Step-by-Step Solution

Answer

By using the shells the volume of cone of radius $3$ and height $5$ is given as :-

$$\begin{gathered} \Rightarrow \mathrm{V}=\frac{2\mathrm{\pi }}{2}{\int }_{-3}^3\left(5\mathrm{x}-\frac{5{\mathrm{x}}^2}{3}\right)\mathrm{dx} \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }{\int }_0^3\left(5\mathrm{x}-\frac{5{\mathrm{x}}^2}{3}\right)\mathrm{dx} \end{gathered}$$

By solving this integration we get $V=15\mathrm{\pi }$. This is the same as the volume by finding using volume formula.

So that the given volume formula $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ holds for cone of radius $3$ and height $5$.

Step 1. Given Information

We have to prove that the volume formula for cone $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ holds for the volume of cone of radius $3$ and height $5$.

Step 2. Volume by using shells method

A cone of radius $3$ and height $5$ can obtained by rotating the region $y=\frac{h}{2}\left(1-\frac{x}{r}\right) or y=\frac{5}{2}\left(1-\frac{x}{3}\right)$ around x-axis on $\left[-3,3\right]$.

Then by using shells method volume is given by :-

$$\begin{gathered} V=\frac{2\mathrm{\pi }}{2}{\int }_{-3}^3\mathrm{x}\left[5\left(1-\frac{\mathrm{x}}{3}\right)\right]\mathrm{dx} \\ \Rightarrow \mathrm{V}=\mathrm{\pi }{\int }_{-3}^3\left(5\mathrm{x}-\frac{5{\mathrm{x}}^2}{3}\right)\mathrm{dx} \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }{\int }_0^3\left(5\mathrm{x}-\frac{5{\mathrm{x}}^2}{3}\right)\mathrm{dx} \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }{\left[\frac{5{\mathrm{x}}^2}{2}-\frac{5{\mathrm{x}}^3}{9}\right]}_0^3 \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\left[\left(\frac{45}{2}-15\right)-\left(0+0\right)\right] \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\left[\frac{45-30}{2}\right] \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\times \frac{15}{2} \\ \Rightarrow \mathrm{V}=15\mathrm{\pi } \end{gathered}$$

Step 3. Volume by using volume formula

The volume of cone is given by the following formula :- 

$V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

Put $r=3,h=5$, then we have :-

$$\begin{gathered} V=\frac{1}{3}\mathrm{\pi }{\left(3\right)}^2\times 5 \\ \Rightarrow \mathrm{V}=15\mathrm{\pi } \end{gathered}$$

So the volume of cone of radius $3$ and height $5$ is $15\mathrm{\pi }$. This is the same as the volume finding by using shells method.

So we can conclude that the given volume formula of cone $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ holds for cone of radius $3$ and height $5$.