Chapter 6.2, Problem 10E

To determine

Find the volume of the solid obtained by rotating the region bounded by the given curves.

Answer to Problem 10E

The volume of the solid obtained by rotating the region bounded by the given curves is $\pi \left[\frac{5}{2}-\frac{1}{2e^4}+\frac{4}{e^2}\right]$

Explanation of Solution

Given information: Given curves $y=e^{-x},y=1,x=2$

Calculation:

Graph the given curves $y=e^{-x},y=1,x=2$

  

When we rotate about $y=2$ , we will get polygon shown in the picture:

  

To find the cross −sectional area by subtract the area of the inner circle from the area of the outer circle:

  

  

Thus, volume of the solid obtained by rotating the region bounded by the given curves is $\pi \left[\frac{5}{2}-\frac{1}{2e^4}+\frac{4}{e^2}\right]$