Q 15. - Calculus

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?

Step-by-Step Solution

Answer

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

Step 1: Given Information

The given $dV$ is volume increment.

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

Step 2: 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.