Example on spherical coord. and trip. integral

[hotcommodity]

[SOLVED] Example on spherical coord. and trip. integral

Homework Statement

Here's the example in the book. They're proving the volume of a sphere using spherical coordinates.

A solid ball T (the region) with constant density $$\delta$$ is bounded by the spherical surface with equation $$\rho = a$$. Use spherical coordinates to compute its volume V.

It says that the bounds are:

$$0 \leq \rho \leq a, 0 \leq \phi \leq \pi, 0 \leq \theta \leq 2 \pi$$

The bounds for $$\phi$$ confuse me. Why does it go from 0 to pi? Wouldn't that only account for half of the sphere?

Any help is appreciated.

 

[Dick]

One angular coordinate measures pole to pole angle, that goes from 0 to pi. The other measures the equatorial angle, that goes 0 to 2pi. Together they cover the whole sphere.

 

[HallsofIvy]

Draw a picture. If $\phi> \pi$ you would be picking up the same points as with $\phi< \pi$, $\theta> \pi$.

 

[hotcommodity]

Oh ok, I get it, as theta goes from 0 to 2pi, phi sweeps the entire sphere. Thank you both.