Spherical vs cylindrical notation

[Calpalned]

Homework Statement

Plotting a point in spherical coordinates means using the format $(\rho, \theta, \phi)$ in place of $(x, y, z)$. Taking a triple integral means replacing $dV$ with $\rho ^2 sin(\phi) d\rho d\theta d\phi$ As you can see, $\rho, \theta, \phi$ are all in the same order.

However, for cylindrical coordinates, my textbook plots $(r, \theta, z)$ for points, but replaces $dV$ with $r dz dr d\theta$. Why are the three integrals suddenly switched around?

Homework Equations

n/a

The Attempt at a Solution

Is there a reason behind this? I dislike rote memorization

 

[blue_leaf77]

Nothing to worry about, $dV$ is an elemental volume with the shape of a "box". You won't care how you are going to arrange the order of the formula for the volume of a box, will you.

 

[Mark44]

I mostly agree with blue_leaf77 -- the order of integration doesn't really matter. The "box" is a rectangular paralellipiped in cartesian coordinates, but in cylindrical or spherical coordinates, the ends aren't flat, and the "box" tapers toward the small end.