Q. 4 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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

provided that this limit exists.

Step 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.