# What's the area of this volume?

1. Jul 24, 2004

### Jin314159

Consider a 3d coordinate system with axis x,y and z.

We are given a circle on the x-z plane with function $$z^2 + (x-a)^2 = a^2$$. We rotate this circle 90 degrees around the z-axis. What's the volume of the resulting surface?

2. Jul 24, 2004

### arildno

Well, shouldn't that be a quarter of a torus?
I'll opt for that and say:
$$V=\pi{a}^{2}*\frac{\pi}{2}a=\frac{\pi^{2}}{2}a^{3}$$

Edit:
The area function for a given angle measured relative to the x-axis (and with the origin as the pole) is $$f(\theta)=\pi{a}^{2}$$

By Cavalieri's principle, we have:
$$V=\int_{0}^{\frac{\pi}{2}}\pi{a}^{2}ad\theta$$

Last edited: Jul 24, 2004