Proof of Volume of a ball

dhlee528

Homework Statement

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

Homework Equations

x = r sin ( phi) cos ( theta)

y = r sin ( phi )sin (theta)

z = r cos ( phi )

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

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