Chapter 6.2, Problem 4E

To determine

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

Answer to Problem 4E

The volume of the solid obtained by rotating the region bounded by the given curves is $\frac{\pi }{2}{e}^4-\frac{\pi }{2}{e}^2$

Explanation of Solution

Given information: Given curves $y=\ln x,y=1,y=2,x=0$

Calculation:

The region is shown in the picture:

  

Rotate about $y-\text{axis}$ we get the solid shown in the picture:

  

When we slice through the point $y$ ,we get a disk with radius $x=e^y$

  

  

Thus, the volume of the solid obtained by rotating the region bounded by the given curves is $\frac{\pi }{2}{e}^4-\frac{\pi }{2}{e}^2$