Q 13. - Calculus

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?

Step-by-Step Solution

Answer

The six forms are $dx dy dz,$ $dx dz dy$, $dy dx dz$, $dy dz dx$, $dz dx dy$, $dz dy dx$.

Step 1: Given Information

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

Step 2: 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 $dx dy dz, dx dz dy, dy dx dz, dy dz dx, dz dy dx, dz dx dy$.

The shape of region determines the order of integration.