Question

Use cylindrical shells to find the volume $V$ of the solid.

A sphere of radius $r$.

Answer 1

Find the volume of the sphere using cylindrical shells:

Consider a circle with equation $x^2+y^2=r^2$.

Then by using the formula of cylindrical shells, the volume of the sphere is given by :

$\begin{array}{rcl}V& =& \pi {\int }_{-r}^r{y}^{2}dx\\ & \Rightarrow & V=2\pi {\int }_0^r{y}^{2}dx\\ & \Rightarrow & V=2\pi {\int }_0^r\left(r^2-x^2\right)dx\\ & \Rightarrow & V=2\mathrm{\pi }{\left[{\mathrm{xr}}^2-\frac{{\mathrm{x}}^3}{3}\right]}_0^{\mathrm{r}}\\ & \Rightarrow & \mathrm{V}=2\mathrm{\pi }\left[{\mathrm{r}}^3-\frac{{\mathrm{r}}^3}{3}-0+0\right]\\ & \Rightarrow & \mathrm{V}=2\mathrm{\pi }\left[\frac{3{\mathrm{r}}^3-{\mathrm{r}}^3}{3}\right]\\ & \Rightarrow & \mathrm{V}=2\mathrm{\pi }\left[\frac{2{\mathrm{r}}^3}{3}\right]\\ & \Rightarrow & \mathrm{V}=\frac{4}{3}{\mathrm{\pi r}}^3\end{array}$

Therefore, the required volume of sphere by using cylindrical shells is $\mathit{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathit{\pi }{\mathit{r}}^{\mathbf{3}}$.