Question

The volume increment$dV=---$ when you use spherical coordinates to evaluate a triple integral. Why is this the standard order of integration for spherical

coordinates?

Short answer

$dV={\rho }^2\sin \varphi d\rho d\theta d\varphi$

Step-by-step solution

Given Information

The given $dV$is volume increment.

The spherical coordinates are$\left(\rho ,\theta ,\varphi \right)$

Simplification

We need to solve it using spherical coordinates to evaluate triple integral.

Mathematically,

$\int \int \int f(\rho ,\theta ,\varphi )dV=\int \int \int f(\rho ,\theta ,\varphi ){\rho }^2\sin \varphi d\rho d\theta d\varphi$

Comparing we get

$dV={\rho }^2\sin \varphi d\rho d\theta d\varphi$

This is because$\rho$is expressed as a function of $\theta$and $\varphi$, it gives the simplest order of integration.

That is why this is the standard order of integration for spherical coordinates.