Use change in variables and iterated integrals theorm to deduce Pappus

[NeoZeon]

1. Homework Statement [/b]

this problem is on page 267 of Advanced calculus of several variables by Edwards, I just can't seem to get a handle on it:

Let aA be a contented set in the right half of the xz plane ,x>0. Define $$\hat{x}$$, the x-coordinates of the centroid of A, by $$\hat{x}=[1/v(A)]\int\int_Axdxdz$$. If $C$ is the set obtained by revolving about the z-axis, that is,$$C=\{(x,y,z)\in R^3:((x^2+y^2)^{1/2},z)\in A)\} $$

then Pappus' theorem asserts that $$v(C) = 2\pi\hat{x}v(A)$$

that is, that the volume of C is the volume of A multiplied by the distance traveled by the centroid of A. Note that C is the image under the cylindrical coordinates map of the set $$B = \{(r,\theta,z)\in R^3:(r,z) \in A, \theta \in [0,2\pi]\}$$

Apply the change of variables and iterated integrals theorems to to deduce Pappus' theorem

The Attempt at a Solution

I am confused about how to get v(A) in the solution. Do I integrate xhat a 3rd time with respect to theta after i change the variables dx and dz into cylindrical coordinates ?

Any hints would be apperciated.

 

[NeoZeon]

solved it.. nvm