# How do you find the volume bounded by #y=ln(x)# and the lines y=0, x=2 revolved about the y-axis?

##### 2 Answers

For the solution by cylindrical shells, see below.

#### Explanation:

Here is a picture of the region and a representative slice taken parallel to the axis of rotation.

The slice is taken at some value of

Revolving about the

The volume of this representative shell is

The radius is shown as a dashed black line in the picture and has length

The height of the shell will be the great

As already mentioned, the thickness is

The representative volume is

Use integration by parts to get

# = 2pi(2ln2-3/4)#

(Rewrite the answer to taste.)

For the solution by washers see below.

#### Explanation:

Here is a picture of the region and a representative slice taken perpendicular to the axis of rotation.

The slice is taken at some value of

The curve

Revolving about the

The volume of this representative washer is

Where

In this case

and

The representative volume is

so

And the volume is

Use integration to get

# = pi[(4ln2-1/2e^(2ln2))-(-1/2e^0)] = pi(4ln2-3/2)#

(Rewrite the answer to taste.)