Q. 64 - Calculus

Question

Use a definite integral to prove that a sphere of radius r has volume given by the formula $V=\frac{4}{3}{\mathrm{\pi r}}^3$. 

Step-by-Step Solution

Answer

It has been proved that sphere has volume $\frac{4}{3}{\mathrm{\pi r}}^3$

Step 1. Given information

The radius of the sphere is $r$

The volume to be proved of the sphere is $V=\frac{4}{3}{\mathrm{\pi r}}^3$

Step 2. Proof of the volume of the sphere.

Draw a diagram of the sphere:

Here a circular slice of thickness $dx$has been cut at a distance of $x$ from the center of the sphere.

The radius of the slice is:

$AB=\sqrt{r^2-x^2}$

The area of the slice is $\mathrm{\pi }{\left(\mathrm{AB}\right)}^2=\mathrm{\pi }({\mathrm{r}}^2-{\mathrm{x}}^2)$

The volume of the slice is $dV=\mathrm{\pi }\left({\mathrm{r}}^2-{\mathrm{x}}^2\right)\mathrm{dx}$

Hence the volume of the sphere is :

$$\begin{gathered} V={\int }_{-r}^r\mathrm{\pi }\left({\mathrm{r}}^2-{\mathrm{x}}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_{-\mathrm{r}}^{\mathrm{r}}\left({\mathrm{r}}^2-{\mathrm{x}}^2\right)\mathrm{dx} \\ =2\mathrm{\pi }{\int }_0^{\mathrm{r}}\left({\mathrm{r}}^2-{\mathrm{x}}^2\right)\mathrm{dx} \\ =2\mathrm{\pi }{\left[{\mathrm{r}}^2\mathrm{x}-\frac{{\mathrm{x}}^3}{3}\right]}_0^{\mathrm{r}} \\ =2\mathrm{\pi }\left({\mathrm{r}}^3-\frac{{\mathrm{r}}^3}{3}-0\right) \\ =2\mathrm{\pi }\left(\frac{2{\mathrm{r}}^3}{3}\right) \\ =\frac{4}{3}{\mathrm{\pi r}}^3 \end{gathered}$$