Q 54 - 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 the graph of $f\left(x\right)=e^x$and the line $y=e$on $\left[0,1\right],$ revolved around the x-axis.

Step-by-Step Solution

Answer

The required volume by using shells is $\mathrm{V}=\frac{\mathrm{\pi }}{2}$

Step 1. Given Information

We have given a function :-

$f\left(x\right)=e^x$.

We have to find the volume of region of graph of this function and  the line $y=e$ on $\left[0,1\right]$, revolved around the x-axis. 

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

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

$V=2\mathrm{\pi }{\int }_{\mathrm{c}}^{\mathrm{d}}\mathrm{r}\left(\mathrm{y}\right)\mathrm{h}\left(\mathrm{y}\right)\mathrm{dy}$

Here axis of revolution is $x-axis$. So that $r\left(y\right)=y$ and the height is $h\left(x\right)=\ln \left(y\right)$.

Then we get the volume as following:-

$$\begin{gathered} V=2\mathrm{\pi }{\int }_0^1\mathrm{y}\left(\ln \left(\mathrm{y}\right)\right)\mathrm{dy} \\ \Rightarrow V=2\mathrm{\pi }{\left[\frac{{\mathrm{y}}^2}{2}\ln \left(\mathrm{y}\right)-\frac{{\mathrm{y}}^2}{4}\right]}_0^1 \\ \Rightarrow V=2\mathrm{\pi }\left[\left(\frac{1}{2}\ln \left(1\right)-\frac{1}{4}\right)-\left(0-0\right)\right] \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\left[0-\frac{1}{4}+0\right] \\ \Rightarrow \mathrm{V}=2\mathrm{\pi }\left[-\frac{1}{4}\right] \\ \Rightarrow \mathrm{V}=-\frac{\mathrm{\pi }}{2} \end{gathered}$$

Also, volume cannot be negative so remove the negative sign, then we have:-

$V=\frac{\mathrm{\pi }}{2}$