Q 65 - Calculus

Question

Prove that a shell with average radius $r$, height $h$, and thickness $∆x$ has volume $V=2\mathrm{\pi rh}∆\mathrm{x}$.

Step-by-Step Solution

Answer

By dividing the shell into different cuboids, we get that the volume of a shell is :-

$V=2\mathrm{\pi rh}∆\mathrm{x}$

Step 1. Given Information

We have to prove that volume of a shell with radius $r$, height $h$ and thickness $∆x$ is $V=2\mathrm{\pi rh}∆\mathrm{x}$.

Step 2. Volume of a shell

We have a shell with

$$\begin{gathered} radius = r, \\ height = h and \\ thickness = ∆x \end{gathered}$$

if we divide the shell into several equal cuboids, then we know that the volume of a cuboid is defined as $length \times breadth \times height$.

So the volume of cuboid will be $rh∆x$.

if we make a complete circle of cuboids, then we get the shell. So the volume of shell will be equals to the $2\mathrm{\pi }$ times the volume of a cuboid as $2\mathrm{\pi }$ is represents a circle.

So the volume of shell is :-

$$\begin{gathered} V=2\mathrm{\pi } \times \mathrm{volume} \mathrm{of} \mathrm{cuboid} \\ \Rightarrow \mathrm{V}=2\mathrm{\pi rh}∆\mathrm{x} \end{gathered}$$

Hence proved.