Chapter 6.3, Problem 13E

To determine

Find the volume enclosed by the given curves.

Answer to Problem 13E

The volume of the solid is $\frac{7}{15}\pi$

Explanation of Solution

Given information: Given curves $y=x^4,y=0,x=1,x=2$

Calculation:

The region is shown in the picture:

  

Rotating with the cylinder method about $x=2$ .

The shaded region with the part that will form the solid.

The height of the each cylinder will be $h=x^4$

The radius will be represented by $r=2-x$

The integrate from $x=0\text{ to }1$

  $\begin{array}{l}V={\displaystyle \underset{a}{\overset{b}{\int }}\left(2\pi rh\right)}dx\\ =2\pi {\displaystyle \underset{0}{\overset{1}{\int }}\left(2-x\right)x^{4}dx}\\ =2\pi {\displaystyle \underset{0}{\overset{1}{\int }}\left(2x^4-x^5\right)dx}\\ =2\pi {\left[\frac{2}{5}{x}^5-\frac{1}{6}{x}^6\right]}_0^1\\ =2\pi \left[\frac{2}{5}-\frac{1}{6}-\left(0-0\right)\right]\\ =2\pi \frac{12-5}{30}\\ =\frac{7}{15}\pi \end{array}$

Thus, the volume of the solid is $\frac{7}{15}\pi$