Q. 5 - Calculus

Question

Explain how to construct a midpoint Riemann sum for a function of three variables over a rectangular solid for which each $\left(x_i^{*},y_j^{*},z_k^{*}\right)$is the midpoint of the subsolid   $R_{ijk}=\left\{\left(x,y,k\right)\right|x_{i-1}\le x_i^{*}\le x_i,y_{j-1}\le y_j^{*}\le y_j and z_{k-1}\le z_k^{*}\le z_k\}$. Refer either to your answer to Exercise $4$ or to Definition $13.14$.

Step-by-Step Solution

Answer

The midpoint of ordinates to use the midpoint Riemann sum is given by,

$\left(x_i^{*},y_j^{*},z_k^{*}\right)=\left(\frac{x_{i-1}+x_i}{2},\frac{y_{j-1}+y_j}{2},\frac{z_{k-1}+z_k}{2}\right)$.

Step 1 . Given information

$R_{ijk}=\left\{\left(x,y,k\right)\right|x_{i-1}\le x_i^{*}\le x_i,y_{j-1}\le y_j^{*}\le y_j and z_{k-1}\le z_k^{*}\le z_k\}$.

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

The midpoint of any three ordinates is given by, $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$.

For each subsolid $R_{ijk}$ choose $\left(x_i^{*},y_j^{*},z_k^{*}\right)=\left(\frac{x_{i-1}+x_i}{2},\frac{y_{j-1}+y_j}{2},\frac{z_{k-1}+z_k}{2}\right)$

Since $x_{i-1}\le x_i^{*}\le x_i,y_{j-1}\le y_j^{*}\le y_j ,z_{k-1}\le z_k^{*}\le z_k$.

This is the midpoint of ordinates to use the midpoint Riemann sum.