# Work, Pappus Theorem

1. Jan 11, 2010

### Pirata

The Region "R" under the graph of y = x^3 from x=0 to x=2 is rotated about the y-axis to form a solid.

a. Find the area of R.
b. Find the volume of the solid using vertical slices.
c. Find the first moment of area of R with respect to the y-axis. What do you notice about the integral?
d. Find the x coordinate of the centroid of R.
e. A theorem of Pappus states that the volume of a solid of revolution equals the area of the region being rotated times the distance the centroid of the region travels. Show that this problem confirms this theorem.

3. The attempt at a solution
I was able to do part "a" as the integral from 0 to 2 of x^3 dx. Also I believe part "b" is pi*[3y^(5/3)/5] evaluated from 0 to 2

2. Jan 11, 2010

### Dick

I don't think "b" is right. How did you get that?

3. Jan 12, 2010

### Pirata

Well I posted what I got. Was hoping you would show me where I went wrong...

4. Jan 12, 2010

### Dick

It's best if you show how you got what you got. Otherwise we just have to guess how you got what you got. That's not fun. Use the method of shells to find the volume. V=integral(2*pi*x*height*dx).