Find The Average Value Of Over The Rectangle With Vertices

Finding the Average Value of a Function Over a Rectangle: A Comprehensive Guide

Finding the average value of a function over a given region is a crucial concept in calculus with applications spanning various fields like physics, engineering, and economics. This article delves into the detailed process of calculating the average value of a function over a rectangular region, providing a comprehensive understanding through examples and explanations.

Understanding the Concept of Average Value

Before diving into the calculations, let's establish a solid understanding of the underlying concept. The average value of a function, f(x,y), over a region R intuitively represents the "average height" of the function's surface above that region. Imagine the function's graph as a landscape; the average value is the height of a flat plane that would enclose the same volume under the surface as the original function over the given area.

Defining the Rectangular Region

We'll focus on finding the average value over a rectangular region. A rectangle in the xy-plane can be defined by its vertices. Let's consider a rectangle with vertices (a, c), (b, c), (a, d), and (b, d), where 'a' and 'b' are the x-coordinates, and 'c' and 'd' are the y-coordinates. This rectangle is defined by the inequalities: a ≤ x ≤ b and c ≤ y ≤ d.

The Formula for Average Value Over a Rectangle

The average value, f_avg, of a continuous function f(x, y) over a rectangular region R defined by a ≤ x ≤ b and c ≤ y ≤ d is given by the double integral:

f_avg = (1/Area(R)) ∬_R f(x, y) dA

Where:

• Area(R) = (b-a)(d-c) represents the area of the rectangle.
• ∬_R f(x, y) dA represents the double integral of the function f(x, y) over the region R. This integral calculates the volume under the surface defined by f(x, y) and above the rectangle R.

Step-by-Step Calculation Process

Let's break down the calculation process into manageable steps:

Step 1: Define the Function and the Region

Clearly define the function f(x, y) and the vertices of the rectangle. This establishes the boundaries for the double integral.

Step 2: Calculate the Area of the Rectangle

Use the formula Area(R) = (b-a)(d-c) to compute the area of the rectangular region.

Step 3: Set up the Double Integral

Set up the double integral based on the function and the region's boundaries:

∬_R f(x, y) dA = ∫_a^b ∫_c^d f(x, y) dy dx (or dx dy, depending on the order of integration)

Step 4: Evaluate the Inner Integral

Evaluate the inner integral with respect to one variable (either 'y' or 'x'), treating the other variable as a constant. This will result in a function of the remaining variable.

Step 5: Evaluate the Outer Integral

Evaluate the outer integral with respect to the remaining variable. The result of this step is a numerical value representing the volume under the surface.

Step 6: Divide by the Area

Finally, divide the result from Step 5 (the volume) by the area calculated in Step 2 to obtain the average value of the function over the rectangle.

Example 1: A Simple Case

Let's find the average value of the function f(x, y) = x + y over the rectangle with vertices (0, 0), (2, 0), (0, 1), and (2, 1).

Step 1: f(x, y) = x + y; a = 0, b = 2, c = 0, d = 1

Step 2: Area(R) = (2 - 0)(1 - 0) = 2

Step 3: ∬_R (x + y) dA = ∫₀² ∫₀¹ (x + y) dy dx

Step 4: Inner integral: ∫₀¹ (x + y) dy = [xy + (y²/2)]₀¹ = x + 1/2

Step 5: Outer integral: ∫₀² (x + 1/2) dx = [(x²/2) + (x/2)]₀² = 3

Step 6: f_avg = 3 / 2 = 1.5

Therefore, the average value of f(x, y) = x + y over the specified rectangle is 1.5.

Example 2: A More Complex Case

Let's find the average value of f(x, y) = x²y over the rectangle with vertices (1, 0), (3, 0), (1, 2), and (3, 2).

Step 1: f(x, y) = x²y; a = 1, b = 3, c = 0, d = 2

Step 2: Area(R) = (3 - 1)(2 - 0) = 4

Step 3: ∬_R x²y dA = ∫₁³ ∫₀² x²y dy dx

Step 4: Inner integral: ∫₀² x²y dy = [(x²y²/2)]₀² = 2x²

Step 5: Outer integral: ∫₁³ 2x² dx = [(2x³/3)]₁³ = 52/3

Step 6: f_avg = (52/3) / 4 = 13/3

Therefore, the average value of f(x, y) = x²y over the specified rectangle is 13/3.

Choosing the Order of Integration

The order of integration (dy dx or dx dy) can significantly impact the complexity of the calculation. Choosing the order that simplifies the integration process is crucial. Sometimes, one order will lead to significantly easier integration than the other.

Handling More Complex Functions

The same principles apply to more complex functions. The key is to carefully evaluate the integrals, possibly using techniques like substitution or integration by parts, to reach the final solution. Remember to always check your work for accuracy.

Applications of Average Value

Calculating the average value of a function has broad applications:

• Physics: Finding the average temperature or pressure over a region.
• Engineering: Determining the average stress or strain in a material.
• Economics: Computing the average revenue or profit over a given time period.
• Image Processing: Calculating the average color intensity in an image region.

Conclusion

Finding the average value of a function over a rectangle is a fundamental concept in multivariable calculus. By following the step-by-step process outlined in this article, along with careful consideration of integration techniques, you can effectively calculate the average value for various functions and rectangular regions. Remember that understanding the underlying concept and choosing the appropriate order of integration are key to efficient and accurate calculations. This knowledge provides a solid foundation for tackling more advanced problems in calculus and its diverse applications.