Chapter 6.2, Problem 24E

To determine

Find the volume enclosed by the given curves.

Answer to Problem 24E

The volume of the solid is $9.82476$

Explanation of Solution

Given information: Given curves $y=x,y=x{e}^{1-\frac{x}{2}};\text{ about }y=3$

Calculation:

  

When we rotate thin vertical strip as shown in figure, about the line $y=3$

We get a washer with inner radius $\left(3-x{e}^{1-\frac{x}{2}}\right)$ and outer radius $\left(3-x\right)$

  $\begin{array}{l}V=\pi {\displaystyle \underset{0}{\overset{2}{\int }}{\left(\text{outerRadius}\right)}^2}-{\left(\text{InnerRadius}\right)}^{2}dx\\ V=\pi {\displaystyle \underset{0}{\overset{\pi }{\int }}{\left(3-x\right)}^2}-{\left(3-x{e}^{1-\frac{x}{2}}\right)}^{2}dx\\ =24e-\frac{142}{3}-2e^2\\ \approx 9.82476\end{array}$

Since the problem is marked $\boxed{\text{CAS}}$.

Thus, the volume of the solid is $9.82476$