Problem 57 - Calculus Volume 3

Question

Show that the average value of a function \(f\) on a rectangular \(\quad\) region \(R=[a, b] \times[c, d]\) is \(f_{\text {ave }} \approx \frac{1}{m n} \sum_{i=1}^{m} \sum_{j=1}^{n} f\left(x_{i j}^{*}, y_{i j}^{*}\right),\) where \(\left(x_{i j}^{*}, y_{i j}^{*}\right)\) are the sample points of the partition of \(R\), where \(1 \leq i \leq m\) and \(1 \leq j \leq n.\)

Step-by-Step Solution

Answer

The average value of the function is approximated by \( f_{\text{ave}} \approx \frac{1}{mn} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*) \).

Step 1: Understand the Problem

We are tasked with finding and proving the average value of a function \( f \) over a rectangular region \( R = [a, b] \times [c, d] \) using sample points \( (x_{ij}^*, y_{ij}^*) \) derived from a partition of the region. We need to show that the approximate average value of the function is given by the expression \( f_{\text{ave}} \approx \frac{1}{mn} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*) \).

Step 2: Define the Partitions of the Region

Partition the rectangular region \( R = [a, b] \times [c, d] \) into \( mn \) smaller rectangles. Assume each rectangle has dimensions \( \Delta x = \frac{b-a}{m} \) and \( \Delta y = \frac{d-c}{n} \), where \( m \) is the number of partitions along the \( x \)-axis and \( n \) is the number along the \( y \)-axis.

Step 3: Calculate the Sample Points

Choose sample points \( (x_{ij}^*, y_{ij}^*) \) within each small rectangle of the partition. These points can be selected as the center or any specific point within each rectangle. These points will be used to evaluate the function \( f \).

Step 4: Set Up the Summation for the Average Value

The approximate average value of \( f \) over the rectangular region can be calculated using the formula:\[f_{\text{ave}} \approx \frac{1}{mn} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*)\]This formula accumulates the function values at each of the \( mn \) sample points and then averages them by dividing by the total number of sample points.

Step 5: Justify the Formula

For a function's average over a continuous region, the integral form is the traditional method. However, when approximating, partitioning the region and taking a Riemann sum of function values at specific sample points achieves an average approximation. \(\frac{1}{mn} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*) \) is the finite sum form mimicking the continuous integration process, and thus provides an approximation of the average value of \( f \) over \( R \).

Key Concepts

Rectangular Region

In mathematics, a rectangular region on the Cartesian plane is often referred to as a domain over which functions or equations are analyzed. This region is defined by two intervals: one along the x-axis and one along the y-axis. For example, consider a region denoted by \( R = [a, b] \times [c, d] \). This specifies that the region spans horizontally from \( a \) to \( b \) and vertically from \( c \) to \( d \).

When discussing an average value of a function over a rectangular region, it's helpful to visualize a grid being formed over this area. Each point within this grid is a potential sample point for the function evaluations. The concept of working within a clearly defined rectangular region helps align with many geometry and calculus applications, creating a pathway for organized analysis.

Partition of a Region

The idea of partitioning is to divide a rectangular region into smaller, more manageable sections or rectangles. This division is crucial in calculus, as it allows for functions to be evaluated over smaller sections, eventually leading to integral approximations.

In practical terms, when you have a rectangular region \( R = [a, b] \times [c, d] \), you partition this region into \( m \) parts along the x-axis and \( n \) parts along the y-axis. Each small rectangle formed has width \( \Delta x = \frac{b-a}{m} \) and height \( \Delta y = \frac{d-c}{n} \). This results in a grid of \( mn \) smaller rectangles, each of which can be used to simplify the evaluation of the function across the entire original region.

Sample Points

Sample points are the specific locations within each partitioned rectangle where the function values are evaluated. These points are essential for setting up the Riemann sum, which approximates the integral over the region.

Choosing sample points \( (x_{ij}^*, y_{ij}^*) \) within each rectangle is a key step. Common selections include the midpoint, left edge, right edge, or any distinguishable point that is consistent throughout the partitions. By evaluating the function at these sample points, you gather approximate values that contribute to estimating the average function value over the rectangular region.

Riemann Sum

A Riemann sum is a method for approximating the total value of a function over a specific region, using a finite sum structure. It's a fundamental concept in calculus for estimating integrals.

The formula \( f_{\text{ave}} \approx \frac{1}{mn} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*) \) mentioned in the exercise encapsulates this concept, where each term \( f(x_{ij}^*, y_{ij}^*) \) represents the function evaluated at a chosen sample point within the partition. The sum of all these values provides an aggregated approximation, and dividing by \( mn \) yields an average. This mimics the behavior of continuous integrals over the total rectangular area but through a discrete approach suitable for computational evaluations.