# Thread: volume of the solid of revolution

1. ## volume of the solid of revolution

region R is bounded by the curve y=ln x, and the lines y=1 and x=1. Find the volume of the solid formed when R is rotated through 360 degree about the y-axis.

What is the graph look like and how to integrate ?

2. Originally Posted by tommylai12
region R is bounded by the curve y=ln x, and the lines y=1 and x=1. Find the volume of the solid formed when R is rotated through 360 degree about the y-axis.

What is the graph look like and how to integrate ?
I am attaching the region in a pdf file.

Use the washer method for this problem, or the shell method if you are familiar with both. Below I quickly discuss the shell method:

V = INTEGRAL ( distance traveled * height of region)

distance = 2 pi x since the radius from the rotational axis to a level x is simply the x coordinate

height of region = 1 - ln x

the bounds are from x = 1 to the intersection point between y=ln x and y=1, which is simply e = 2.71828...

So the volume is INTEGRAL (2pi x (1-ln x)) from 1 to e.

The washer method, briefly:

Outer radius R(y) = x = e^y
Bounds y = 0 to y =1

Formula: V = INTEGRAL (Pi R^2 - Pi r^2)

Good luck!!

3. Thanks for your help. It helps a lot

4. You're welcome. The answer turns out to be: 6.894313198.

Attached find a 3D image of the solid. The horizontal line extending from left to right on the green surface is y = 1, and the curve to the right (concave down) is the graph of ln (x).

5. Originally Posted by tommylai12
region R is bounded by the curve y=ln x, and the lines y=1 and x=1. Find the volume of the solid formed when R is rotated through 360 degree about the y-axis.

What is the graph look like and how to integrate ?
Volume=Pi*int{(e^y)^2-1^1)}[/tex]

The region:

6. Originally Posted by apcalculus
You're welcome. The answer turns out to be: 6.894313198.

Attached find a 3D image of the solid. The horizontal line extending from left to right on the green surface is y = 1, and the curve to the right (concave down) is the graph of ln (x).

So that mean integrate lnx from 0 to 1?

I know already... INTEGRAL (2pi x (1-ln x)) from 1 to e...thanks.