Linear algebra and volume of rotational object

[Niles]

[SOLVED] Linear algebra and volume of rotational object

Homework Statement

I have to show that V(O) = pi * int[f(z)^2]dx when O = {(x,y,z) E R^3 | a =< z =< b, sqrt(x^2+y^2) =< f(z)}.

I have to integrate 3 times with different limits:

V(O) = int [dz, a..b] * int[dr, 0 .. f(z)] * int[dTheta * r, 0..2pi].

Why is it the integral looks like this? I believe it should look like

int[ int [ int [ dr * dTheta * ds]]] with limits as above.

 

[HallsofIvy]

You have a< z< b, with a and b constants, so the "outer integral" would be with respect to z, from a to b, in order to cover the entire figure. For each z, you have $\sqrt{x^2+ y^2}< f(z)$. Taking $r= \sqrt{x^2+ y^2}$, that is the same as r< f(z). Since r (in polar coordinates) cannot be less than 0, the limits of integration with respect to r must be from 0 to f(z) (and the differential is rdr). Finally, there is no limit on $\theta$ so the limits on $\theta$ must be from 0 to $\pi$, the entire circle. That volume is given by
$$\int_{z=a}^{b} \int_{r= 0}^{f(x)} \int_{\theta= 0}^{2\pi} f(z) (d\theta)(rdr)(dz)$$
It should be easy to see that gives the result you want.

As for the formula you give, I see no difference except that you have replaced "z" with "s"- without saying what "s" is.

 

[Niles]

Why is r^2 = x^2+y^2? And isn't this cylindrical coordinates?

Thanks for your help so far!