# Proof of Volume of a ball

1. Mar 3, 2010

### dhlee528

1. The problem statement, all variables and given/known data

http://staff.washington.edu/dhlee528/003.JPG [Broken]

2. Relevant equations

x = r sin ( phi) cos ( theta)

y = r sin ( phi )sin (theta)

z = r cos ( phi )

3. The attempt at a solution

$$vol=8 \int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2}\int_0^r \rho^2 \sin(\phi)d\rho d\theta d\phi$$

$$8 \int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2} \sin(\phi)(\frac{\rho^3}{3}){|}_0^r d\theta d\phi$$

$$\frac{4r^3 \pi}{3}\int_0^\frac{\pi}{2}sin(\phi)d\phi$$

$$-\frac{4r^3\pi}{3}[0-1]=\frac{4\pi r^3}{3}$$

I think I got spherical coordinate right but don't know how to do for rectangular or spherical coordinate

Last edited by a moderator: May 4, 2017
2. Mar 3, 2010

### tiny-tim

Welcome to PF!

Hi dhlee528! Welcome to PF!

(have a theta: θ and a phi: φ and a pi: π )

For rectangular coordinates: obviouly the volume element is dxdydz, so decide which order you're going to integrate in … say keep z and y fixed, decide the limits on x; then keep z fixed, decide the limits on y.

What do you get?