Problem 9 - 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. \(f(x, y)=12-x-y\)

Step-by-Step Solution

Answer

The approximate Riemann sum is 168.

Step 1: Identify the Region and Partition

The region \( R \) can be divided into six equal squares using the lines \( x = 2 \), \( x = 4 \), and \( y = 2 \). This results in three vertical and two horizontal partitions, creating six squares.

Step 2: Calculate Center of Each Square

Each square has a unique center, described as follows: \((1, 1)\), \((3, 1)\), \((5, 1)\), \((1, 3)\), \((3, 3)\), \((5, 3)\). These are midpoints of each square, acting as sample points for the Riemann sum.

Step 3: Determine Area of Each Square

Since the whole region is divided into six squares, and each line separates the region at every two units, each square has side length of 2 units. Thus, the area \( \Delta A_{k} \) for each square is \( (2 \times 2) = 4 \).

Step 4: Evaluate Function at Each Center

For each center \( (\bar{x}_k, \bar{y}_k) \), we evaluate \( f(x, y) = 12 - x - y \):- \( f(1, 1) = 10 \)- \( f(3, 1) = 8 \)- \( f(5, 1) = 6 \)- \( f(1, 3) = 8 \)- \( f(3, 3) = 6 \)- \( f(5, 3) = 4 \)

Step 5: Calculate the Riemann Sum

Calculate the Riemann sum by adding up the function values at the centers of the squares, each multiplied by the area of the square:\[\sum_{k=1}^{6} f(\bar{x}_{k}, \bar{y}_{k}) \Delta A_{k} = 10 \times 4 + 8 \times 4 + 6 \times 4 + 8 \times 4 + 6 \times 4 + 4 \times 4 = 168 \]

Key Concepts

Partitioning a Region

In calculus, partitioning a region is a key step in exploring the concept of Riemann sums and integrals. It involves dividing a complex area into smaller, more manageable shapes. In our exercise, the region is neatly partitioned into six equal squares. This is achieved using vertical lines at \( x = 2 \) and \( x = 4 \), and a horizontal line at \( y = 2 \). By making these divisions:

• We create a grid of smaller regions within the larger area.
• This simplification makes calculating the integral more approachable.

Understanding partitioning helps us visualize how complex shapes can be broken down into simpler parts, allowing us to approximate or calculate their area using Riemann sums.

Function Evaluation

Function evaluation is a fundamental step in working with Riemann sums. It involves determining the function values at specific points within each subdivided region. For the given function \( f(x, y) = 12 - x - y \), the goal is to compute the value of \( f \) at the center of each square. These centers in our example are:

• \((1, 1)\), where \( f(1, 1) = 10 \)
• \((3, 1)\), where \( f(3, 1) = 8 \)
• \((5, 1)\), where \( f(5, 1) = 6 \)
• \((1, 3)\), where \( f(1, 3) = 8 \)
• \((3, 3)\), where \( f(3, 3) = 6 \)
• \((5, 3)\), where \( f(5, 3) = 4 \)

Evaluating the function at these points gives us the necessary data to approximate the integral over the entire region using the Riemann sum.

Approximation Method

The approximation method employed in this exercise is the Riemann sum, which approximates the integral of a function over a region. To do this:

• We calculate the function value at each partition's center, as this provides a sample point.
• Multiply each function value by the area of its corresponding square; in this scenario, each square has an area of \(4\).
• Sum all these products to get an approximation of the integral over the entire region.

In our example, the Riemann sum is calculated as follows:\[ \sum_{k=1}^{6} f(\bar{x}_{k}, \bar{y}_{k}) \Delta A_{k} = 10 \times 4 + 8 \times 4 + 6 \times 4 + 8 \times 4 + 6 \times 4 + 4 \times 4 = 168 \]This sum approximates the double integral of the function over the given area. Riemann sums enable us to estimate integrals using a finite series of calculations, making them a powerful tool in analysis and calculus.