triple integral transformation

[cheffy]

We learned how to find the volume of a solid of revolution using the shell method; namely, if the region between the curve y=f(x) and the x=axis from a to b (0<a<b) is revolved about the y-axis, the volume of the resulting solid is
\(\displaystyle \int_a^b {2\pi xf(x)dx}\)
Prove that finding volumes by using triple integrals gives the same result. (hint: use cylindrical coordinates with the roles of y and z changed).

Help! I don't understand what this is telling me. >_<

 

[galactus]

Are you familiar with the Theorem of Pappus?.

 

[cheffy]

Are you familiar with the Theorem of Pappus?.

It's not ringing any bells, at least under that name.

 

[galactus]

It should be in your calc book(assuming you have one). Or Google it. There should be lots on it. It's actually pretty cool.

It is

\(\displaystyle \text{volume = (area of region)(distance travelled by centroid)}\)

By Pappus' theorem:

\(\displaystyle V=\sum_{k=1}^{n}2{\pi}x_{k}{\Delta}A_{k}\)

\(\displaystyle V=\int\int2{\pi}xdA\)

\(\displaystyle =2{\pi}\int\int{x}dA\)

\(\displaystyle V=2{\pi}\overline{x}[\text{area of region}]\)

It is very similar to shells method.

A triple integral could be implemented by including the area of the region

That's along the line in which I was thinking anyway.

 

[cheffy]

How would I set up the bounds of the triple integral?

And should that hint they gave in the question mean anything?