Q 61 - Calculus

Question

Use definite integrals to find the volume of each solid of revolution described in Exercises $49-61$. (It is your choice whether to use disks/washers or shells in these exercises.) 

The region bounded by the graphs of $f\left(x\right)=\left|x\right|$ and $y=2$ on $\left[-2,2\right]$, revolved around the line $x=3$. 

Step-by-Step Solution

Answer

The required volume by using shells is $V=24\mathrm{\pi }$

Step 1. Given Information

We have given the following function :-

$f\left(x\right)=\left|x\right|$.

We have to find the volume of region of graph of this function and $y=2$

Step 2. Find the integral and evaluate it to calculate volume

We know that by using shells the volume is given by :-

$V=2\mathrm{\pi }{\int }_{\mathrm{d}}^{\mathrm{d}}\mathrm{r}\left(\mathrm{x}\right)\mathrm{h}\left(\mathrm{x}\right)\mathrm{dx}$.

Here axis of revolution is $x=3$. So that $r\left(x\right)=3-x$.

Also from $y=-2 to 0$ height is given by $h\left(x\right)=2+x$

and from $y=0 to 2$ height is given by $h\left(x\right)=2-x$.

So the volume is given by following :-

$$\begin{gathered} V=2\mathrm{\pi }{\int }_{-2}^0\left(3-\mathrm{x}\right)\left(2+\mathrm{x}\right)\mathrm{dx}+2\mathrm{\pi }{\int }_0^2\left(3-\mathrm{x}\right)\left(2-\mathrm{x}\right)\mathrm{dx} \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }{\int }_{-2}^0\left(6+\mathrm{x}-{\mathrm{x}}^2\right)\mathrm{dx}+2\mathrm{\pi }{\int }_0^2\left(6-5\mathrm{x}+{\mathrm{x}}^2\right)\mathrm{dx} \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }{\left[6\mathrm{x}+\frac{{\mathrm{x}}^2}{2}-\frac{{\mathrm{x}}^3}{3}\right]}_{-2}^0+2\mathrm{\pi }{\left[6\mathrm{x}-\frac{5{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^3}{3}\right]}_0^2 \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\left[\left(0+0-0\right)-\left(-12+2+\frac{8}{3}\right)\right]+2\mathrm{\pi }\left[\left(12-10+\frac{8}{3}\right)\right] \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\left[12-2-\frac{8}{3}+12-10+\frac{8}{3}\right] \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\left[12\right] \\ \Rightarrow \mathrm{V}=24\mathrm{\pi } \end{gathered}$$