Problem 58 - Calculus Volume 3

Question

Use the midpoint rule with \(m=n\) to show that the average value of a function \(f\) on a rectangular region \(R=[a, b] \times[c, d]\) is approximated by $$ f_{\text {ave }} \approx \frac{1}{n^{2}} \sum_{i, j=1}^{n} f\left(\frac{1}{2}\left(x_{i-1}+x_{i}\right), \frac{1}{2}\left(y_{j-1}+y_{j}\right)\right). $$

Step-by-Step Solution

Answer

The average value is approximated by evaluating the function at midpoints and dividing by the total number of rectangles, \( n^2 \).

Step 1: Understand the Midpoint Rule

The midpoint rule is used for approximating the average value of a function over a region. It involves evaluating the function at the midpoints of subintervals created by dividing the region into smaller rectangles.

Step 2: Divide the Region into Subintervals

Divide the region \( R = [a, b] \times [c, d] \) into \( n \times n \) smaller subrectangles. The dimensions of each subrectangle are \( \Delta x = \frac{b-a}{n} \) and \( \Delta y = \frac{d-c}{n} \).

Step 3: Determine the Midpoints

Identify the midpoints for each subrectangle. For the midpoint in the \(i\)-th and \(j\)-th subrectangle, use the coordinates: \( x_i = \frac{1}{2}(x_{i-1} + x_i) \) and \( y_j = \frac{1}{2}(y_{j-1} + y_j) \).

Step 4: Approximate the Sum Using Midpoints

Calculate the approximate sum of the function values at these midpoints: \[\sum_{i=1}^{n} \sum_{j=1}^{n} f\left( \frac{1}{2} (x_{i-1} + x_i), \frac{1}{2} (y_{j-1} + y_j) \right)\]

Step 5: Compute the Average Value

The average value of the function over the region is the sum of the function values at all midpoints divided by the total number of subrectangles. This is given by:\[f_{\text{ave}} \approx \frac{1}{n^2} \sum_{i,j=1}^{n} f\left(\frac{1}{2}(x_{i-1}+x_i), \frac{1}{2}(y_{j-1}+y_j)\right)\]

Step 6: Conclusion: Expression Verification

By applying the midpoint rule to all \( n^2 \) subrectangles, we have derived the required approximation for the average value of \( f \) over \( R \) using the midpoints. The formula is indeed consistent with the midpoint rule for two-dimensional integration.

Key Concepts

Average Value of a Function

The concept of the average value of a function, especially over a specific region, is crucial in calculus. When dealing with functions over a rectangular region, the average value provides a simplified representation of the entire function's behavior throughout that space.
The average value, denoted as \(f_{\text{ave}}\), is essentially the "central" value of the function within the region. It represents what the function value would be if it were constant throughout the region.
To compute this for a rectangular region, instead of relying on exact calculations (which might be complex), we use approximations. The Midpoint Rule comes handy here.

• This rule simplifies calculations by estimating the average function value based on the function's behavior at the midpoints of subrectangles.
• The approximation formula involves summing the function’s values at these midpoints and dividing them by the number of subrectangles.

By employing this method, we can estimate the average value efficiently.

Rectangular Region

In two-dimensional calculus, a rectangular region is a basic geometric shape that forms the domain over which functions are often evaluated. It's defined by two intervals, one along each axis, namely the x-axis and y-axis, creating a means to analyze functions easily.
A rectangular region is denoted as \(R = [a, b] \times [c, d]\) in mathematical notation. This indicates a stretch from \(a\) to \(b\) along the x-axis and from \(c\) to \(d\) along the y-axis.
Such simplicity allows for easier division into smaller parts, or subrectangles, which aids in approximation techniques such as the Midpoint Rule.

• This division is key to simplifying computations by focusing on smaller, manageable areas.
• Working within this framework helps in calculating aspects like the average value of a function over the given region.

Thus, recognizing and working with rectangular regions is foundational to understanding applied calculus methods.

Subrectangles

Dividing a rectangular region into subrectangles is an important step in numerical integration techniques. The idea is to split the larger, more complex region into smaller, manageable pieces where calculations can be more easily handled.
When dividing a rectangular region \(R = [a, b] \times [c, d]\), it is common to divide it into \(n \times n\) equal subrectangles.
Each subrectangle has dimensions \(\Delta x = \frac{b-a}{n}\) and \(\Delta y = \frac{d-c}{n}\).

• This division creates a grid of smaller rectangles, known as subrectangles, throughout the region.
• Each of these subrectangles becomes a focal point for evaluating the function at its midpoint.

Using subrectangles allows for a systematic approach to numerical integration by simplifying the task into repeated evaluations over these smaller areas.

Two-Dimensional Integration

Two-dimensional integration involves extending basic integration concepts to functions of two variables within a defined region, like a rectangle. It essentially sums over the region by considering both dimensions to calculate areas or volumes.
This approach is crucial for functions that depend on two varying quantities. One common method of approximating these integrals is using the Midpoint Rule.
In the two-dimensional Midpoint Rule, the function is evaluated at the center, or midpoint, of each subrectangle within the region.

• For each subrectangle, the midpoint provides a single value that represents the function’s behavior throughout that subregion.
• By summing the function's values at all midpoints and averaging them, we achieve an approximation of the integral over the entire rectangular region.

The two-dimensional integration simplifies complex operations, making it invaluable for practical calculations in physics and engineering.