Q. 11 - Calculus

Question

Explain how to construct a midpoint Riemann sum for a function of two variables over a rectangular region for which each $\left(x_j^{*},y_k^{*}\right)$ is the midpoint of the subrectangle

${\mathcal{R}}_{\mathbf{j}\mathbf{k}}\mathbf{=}\left\{(x,y)\mid x_{j-1}\le x_j^{*}\le x_j\text{ and }{y}_{k-1}\le y_k^{*}\le y_k\right\}\mathbf{.}$

Refer to your answer to Exercise 10 or to Definition 13.3.

Step-by-Step Solution

Answer

Ans:

Step 1: Subdivide the region

Divide the rectangular region into \( m \times n \) subrectangles \( R_{jk} \).

Step 2: Choose midpoints and sum

For each subrectangle, choose the midpoint \( (x_j^*, y_k^*) \). The midpoint Riemann sum is \( \sum_{j=1}^m \sum_{k=1}^n f(x_j^*, y_k^*) \Delta A \), where \( \Delta A \) is the area of each subrectangle.