Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

The partition of \(R\) into six equal squares by the lines \(x=2, x=4\), and \(y=2\). Approximate \(\iint_{R} f(x, y) d A\) by calculating the corresponding Riemann sum \(\sum_{k=1}^{6} f\left(\bar{x}_{k}, \bar{y}_{k}\right) \Delta A_{k}\), assuming that \(\left(\bar{x}_{k}, \bar{y}_{k}\right)\) are the centers of the six squares. \(f(x, y)=e^{x y}\)

Step-by-Step Solution

Answer

The Riemann sum is approximated by \( 4(e^1 + e^3 + e^5 + e^3 + e^9 + e^{15}) \).

Step 1: Define the Region and Partition

The region \( R \) is partitioned by the vertical lines \( x=2, x=4 \) and the horizontal line \( y=2 \). It creates 6 equal squares within \( R=[0,6] \times [0,4] \). Each square has dimensions \( ext{width} = 2 \) and \( ext{height} = 2 \).

Step 2: Identify Subrectangles and Their Midpoints

The six squares are bounded by: \([0,2]\times[0,2]\), \([2,4]\times[0,2]\), \([4,6]\times[0,2]\), \([0,2]\times[2,4]\), \([2,4]\times[2,4]\), \([4,6]\times[2,4]\). The centers (\( \bar{x}_k, \bar{y}_k \)) are calculated as: \((1,1), (3,1), (5,1), (1,3), (3,3), (5,3)\).

Step 3: Calculate Area of Each Subrectangle

The area \( \Delta A_k \) of each square is \( 2 \times 2 = 4 \).

Step 4: Compute Riemann Sum Contributions

Evaluate \( f(x,y) = e^{xy} \) at each midpoint: \[ f(1,1) = e^1, \ f(3,1) = e^3, \ f(5,1) = e^5, \ f(1,3) = e^3, \ f(3,3) = e^9, \ f(5,3) = e^{15} \].

Step 5: Sum All Contributions to the Riemann Sum

Compute the Riemann sum: \[ \sum_{k=1}^{6} f(\bar{x}_k, \bar{y}_k) \Delta A_k = 4(e^1 + e^3 + e^5 + e^3 + e^9 + e^{15}) \].

Step 6: Final Calculation

Calculate the final sums: \[ 4(e^1 + e^3 + e^5 + e^3 + e^9 + e^{15}) \]. Use a calculator for numerical approximation if necessary.

Key Concepts

Double Integral

A double integral is used to find the volume under a surface in a two-dimensional area. This can be thought of as extending the idea of a Riemann integral to functions of two variables over a region in the plane. In mathematical terms, the double integral of a function \( f(x,y) \) over a region \( R \) is denoted by \( \iint_{R} f(x, y) \, dA \). Here, \( R \) represents the region over which the integral is being evaluated.
This type of integral can calculate a wide range of quantities, such as areas, volumes, and averages. The process involves partitioning the region into subregions, calculating the function's value at certain points in these subregions, and summing all these contributions together. It provides a more comprehensive view of how a function behaves over an area rather than just along a line.

Partition of a Region

Partitioning a region is an essential step in calculating a Riemann sum or a double integral. It involves dividing the entire region into smaller and manageable pieces, like squares or rectangles. In the given exercise, this step included sectioning the region \( R \) into six equal squares using the lines \( x=2, x=4 \), and \( y=2 \). This resulted in a structured and organized grid that allowed easy evaluation of the function at specific subregions.
The objective is to simplify the problem by focusing on smaller segments, where calculations can be handled more easily. Each segment is considered a subrectangle of the larger region, and its properties, such as its midpoint and area, become crucial for further computations. This deconstruction of the region is the foundation upon which the approximation of the integral is built.

Midpoint Rule

The midpoint rule is a method used to approximate the value of an integral, particularly in the context of Riemann sums. In this technique, the function's value is estimated at the midpoint of each partitioned subregion. For the given exercise, this involved finding the center of each of the six squares created by the regional partition. The midpoints are calculated as \( (1,1), (3,1), (5,1), (1,3), (3,3), \text{and} (5,3) \).
By using the midpoint, the method aims to provide a better approximation than just randomly choosing any point or the endpoints. This is because the function value at the midpoint typically averages the values at the endpoints, balancing out over- and underestimation. The chosen midpoints allow the calculation of \( f(\bar{x}_k, \bar{y}_k) \) for each subrectangle, playing a crucial role in forming the Riemann sum to approximate the double integral.

Exponential Function

An exponential function is a mathematical function of the form \( f(x) = e^{xy} \), where \( e \) is Euler's number, approximately equal to 2.71828. These functions are significant due to their constant rate of growth or decay, appearing frequently in various natural processes and scientific models.
In the context of this exercise, the exponential function \( f(x, y) = e^{xy} \) was evaluated at the midpoints of each partitioned square to determine the Riemann sum. Each evaluation involves plugging the midpoint coordinates into the function, leading to expressions like \( e^1, e^3, e^5, e^9, \text{and} e^{15} \).

• Such functions have distinct characteristics, such as continuous growth and never crossing the x-axis.
• The exponential nature means the outputs can increase rapidly, highlighting their impact in calculations.

Understanding these functions is crucial to appreciating how they contribute to the overall Riemann sum and the double integral approximation.