   Chapter 7.1, Problem 65E

Chapter
Section
Textbook Problem

# Calculate the volume generated by rotating the region bounded by the curves y = ln x , y = 0 and x = 2 about each axis.(a) The y-axis(b) The x-axis

To determine

(a)

To evaluate: the volume generated by rotating the region bounded by the given curves about the y axis using the cylindrical shell method.

Explanation

Consider a function y=f(x) between the points x=a and x=b. If the region under the curve y=f(x) between a and b is rotated about the y-axis, a solid shape is obtained. The volume of that solid shape can be calculated by taking the volume of small cylindrical shells making up the solid and then adding them together.

Formula used:

Volume of the solid obtained by rotating the region from a to b under the curve y=f(x) is given by the following integral:

If the curve is rotated about the x-axis, then the volume will be

V=f1(a)f1(b)2πyf1(y)dy,

Given:

The curves,y=lnx

Bounded by y=0,x=2

Calculation:

The region being rotated is as shown below:

When y=0, the x value of the curve lnx will be 1. So, the bounds on x-axis for the region rotated is from 1 to 2.

Substitute the curves into the volume formula with a and bas 1 and 2 respectively;

V=122πxlnxdx=2π12xlnxdx …… (1)

Solve the integral 12xlnxdx using integration by parts. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate

To determine

(b)

To evaluate: the volume generated by rotating the region bounded by the given curves about the x axis using the cylindrical shell method.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Find more solutions based on key concepts 