Problem 74 - Thomas Calculus

Question

Approximate the double integral of \(f(x, y)\) over the region \(R\) partitioned by the given vertical lines \(x=a\) and horizontal lines \(y=c .\) In each subrectangle, use \(\left(x_{k}, y_{k}\right)\) as indicated for your approximation. \begin{equation}\iint_{R} f(x, y) d A \approx \sum_{k=1}^{n} f\left(x_{k}, y_{k}\right) \Delta A_{k} \end{equation} \(f(x, y)=x+2 y\) over the region \(R\) inside the circle \((x-2)^{2}+(y-3)^{2}=1\) using the partition \(x=1,3 / 2,2,5 / 2,\)3 and \(y=2,5 / 2,3,7 / 2,4\) with \(\left(x_{k}, y_{k}\right)\) the center (centroid) in the \(k\) th subrectangle (provided the subrectangle lies within \(R )\)

Step-by-Step Solution

Answer

The approximate integral is the sum of \(f(x_k, y_k) \Delta A_k\) for subrectangles within the circle.

Step 1: Understand the Region R

The region \( R \) specified in the problem is inside the circle \((x-2)^2 + (y-3)^2 = 1\). This represents a circle centered at \((2, 3)\) with radius \(1\). All partitioned subrectangles should lie within this circle.

Step 2: List the Partition Lines

Partition lines given are \(x=1, \frac{3}{2}, 2, \frac{5}{2}, 3\) and \(y=2, \frac{5}{2}, 3, \frac{7}{2}, 4\). We will calculate the center for each subrectangle and check if it lies inside \(R\).

Step 3: Calculate Areas of Subrectangles

Each subrectangle formed from consecutive partition lines \((x_i, x_{i+1})\) and \((y_j, y_{j+1})\) has area \(\Delta A_k = (x_{i+1} - x_i) \times (y_{j+1} - y_j)\). Compute these for subrectangles in domain.

Step 4: Determine Centroid of Each Subrectangle

Find the centroid \((x_k, y_k)\) of each subrectangle, given by \(x_k = \frac{x_i + x_{i+1}}{2}\) and \(y_k = \frac{y_j + y_{j+1}}{2}\). Verify if each centroid lies within the circle \((x-2)^2 + (y-3)^2 < 1\).

Step 5: Evaluate Function at Each Valid Centroid

For each valid centroid \((x_k, y_k)\) within the specified region, evaluate the function \(f(x_k, y_k) = x_k + 2y_k\).

Step 6: Sum Up the Approximate Integral

Sum the products \(f(x_k, y_k) \Delta A_k\) for all subrectangles whose centers lie within \(R\): \[\iint_R f(x,y) \, dA \approx \sum_{k} f(x_k, y_k) \Delta A_k \]

Key Concepts

Partitioning of Regions

The process of partitioning a region involves dividing a larger area, in this case, the circular region defined by \((x-2)^2 + (y-3)^2 = 1\), into smaller, manageable sections or subrectangles. This step is crucial as it facilitates the approximation of the double integral over the region. To partition the region, lines are drawn vertically at \(x=1, 3/2, 2, 5/2, \text{and} 3\), and horizontally at \(y=2, 5/2, 3, 7/2, \text{and} 4\) to create a grid. This grid helps to identify precise areas, thereby simplifying the calculation process that follows. Keep in mind that not all subrectangles formed by these partitions will fall completely within the circle, so verifying each segment is necessary for accurate interpretation.

Centroid of Subrectangles

Once the region is partitioned into subrectangles, the next key step is to locate the centroid of each subrectangle. The centroid, often referred to as the geometric center, is the point \((x_k, y_k)\) that represents an average position of the subrectangle. The centroid is calculated using the formulas:

• \(x_k = \frac{x_i + x_{i+1}}{2}\)
• \(y_k = \frac{y_j + y_{j+1}}{2}\)

These coordinates must then be assessed to ensure that the centroid lies within the boundary of the initial circular region \((x-2)^2 + (y-3)^2 = 1\). Only those centroids which fall inside this area are valid for further computations.

Area Calculations

Determining the area of each subrectangle is a pivotal part of the approximation process. The area \(\Delta A_k\) of any subrectangle can be swiftly calculated using:

• \((x_{i+1} - x_i) \times (y_{j+1} - y_j)\)

This formula ensures you compute the width and height, multiplying them gives the precise area of the subrectangle. These calculated areas play a critical role when combining them with the function evaluations to estimate the integral over the complex shape of the original region.

Validity of Centroids

After establishing centroids for each subrectangle, confirming their validity is the next important task. A valid centroid is situated within the constraints of the circular boundary \((x-2)^2 + (y-3)^2 < 1\). This typically involves substituting the coordinates of each centroid into the circle's equation. If the result shows the centroid lies inside the circle, it is deemed valid. This crucial check ensures that any calculations, such as evaluating the function at these centroids, are only performed on points that legitimately lie within the specified region, thereby avoiding inaccurate extrapolation of results.

Evaluating Functions at Centroids

Finally, the process of approximating the double integral involves evaluating the function at each valid centroid. The given function, \(f(x, y) = x + 2y\), needs to be computed at each valid \((x_k, y_k)\). Each function evaluation gives rise to a product:

• \(f(x_k, y_k) \Delta A_k\)

This product is calculated for each subrectangle whose centroid is validated. The aggregation, or sum, of these products then approximates the double integral, providing an estimate of the overall sum of \(f(x, y)\) over the entire enclosed region. This method is not only efficient but offers a systematic approach to tackle complex integrals by breaking the problem into simpler, manageable parts.