Question

What are the six forms used to express the volume increment $dV$when you use rectangular coordinates to evaluate a triple integral? How do you decide which order to use?

Short answer

The six forms are $dxdydz,$$dxdzdy$, $dydxdz$, $dydzdx$, $dzdxdy$, $dzdydx$.

Step-by-step solution

Given Information

It is given that we need to evaluate$dV$in rectangular system as triple integral.

Simplification

Let $\left(x,y,z\right)$be the coordinates in rectangular coordinate system.

Let $dx,dy,dz$be small increment.

The volume in rectangular coordinates is given by $dV=dxdydz$

The six forms used to express the volume increment $dV$will be $dxdydz,dxdzdy,dydxdz,dydzdx,dzdydx,dzdxdy$.

The shape of region determines the order of integration.