Question

Explain how to construct a Riemann sum for a function of three variables over a rectangular solid.

Short answer

The triple integral of $f$over $R$is given by the following Riemann Sum:

role="math" localid="1650345492114" $\underset{R}{\iiint }f\left(x,y,z\right)dV=\underset{l,m,n\to \infty }{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f\left(x_i,y_j,z_k\right)∆V$

provided that this limit exists.

Step-by-step solution

Step 1. Given information

We need to explain how to construct a Riemann sum for a function of

three variables over a rectangular solid.

Step 2.Explanation for constructing a Riemann sum for a function of three variables over a rectangular solid.

Let $a_1<a_2,b_1<b_2$and $c_1<c_2$be real numbers and $R$be the rectangular solid defined by,

$R=\left\{\left(x,y,z\right)\right|a_1\le x\le a_2,b_1\le y\le b_2$and $c_1\le z\le c_2\}$

Let $f\left(x,y,z\right)$be a function defined on $R$.

The triple integral of $f$over $R$is given by the following Riemann sum:

$$\begin{gathered} \underset{R}{\iiint }f\left(x,y,z\right)dV=\underset{max∆x_i,∆y_j,∆z_k\to 0}{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f\left({x^{*}}_i,{y^{*}}_j,{z^{*}}_k\right)∆V_{ijk} \\ =\underset{l,m,n\to \infty }{\lim }\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f\left(x_i,y_j,z_k\right)∆V \end{gathered}$$

provided that this limit exists.