https://file.aiquickdraw.com/mathbot/file/1724690160153vkjpxg14.jpg

The problem asks to estimate the integral \(\int_R g(x,y) \, dA\) using the midpoint rule, where \(R = [3,5] \times [4,8]\). The table of values for \(g(x, y)\) is provided.

### Step 1: Identify the relevant values
The region \(R\) is a rectangular area in the \(xy\)-plane with \(x\) ranging from 3 to 5 and \(y\) ranging from 4 to 8. The table provides the values of \(g(x, y)\) at specific points. The values relevant to the region \(R\) are:

\[
\begin{array}{c|cccc}
x \backslash y & 4 & 5 & 6 & 7 & 8 \\
\hline
3 & 2 & 0 & 3 & 6 & 7 \\
4 & 4 & 3 & 0 & 2 & 5 \\
5 & 6 & 1 & 3 & 4 & 6 \\
\end{array}
\]

### Step 2: Apply the Midpoint Rule
The Midpoint Rule for a double integral is given by:

\[
\int_R g(x, y) \, dA \approx \sum_{i,j} g(x_i^*, y_j^*) \Delta x \Delta y
\]

Here, \(\Delta x\) and \(\Delta y\) are the widths of the subintervals in the \(x\) and \(y\) directions respectively, and \(g(x_i^*, y_j^*)\) are the values of the function at the midpoints of the subrectangles.

For this table, each subrectangle has \(\Delta x = 1\) and \(\Delta y = 1\). The midpoints in \(x\) will be at 3.5 and 4.5, and in \(y\) at 4.5, 5.5, 6.5, and 7.5. 

The corresponding values of \(g(x_i^*, y_j^*)\) are:

\[
\begin{array}{c|cccc}
(x_i^*, y_j^*) & 4.5 & 5.5 & 6.5 & 7.5 \\
\hline
(3.5, y) & 1 & 1.5 & 3 & 5.5 \\
(4.5, y) & 4.5 & 2 & 1 & 2.5 \\
\end{array}
\]

### Step 3: Calculate the Midpoint Rule Approximation

\[
\int_R g(x, y) \, dA \approx (1 \times 1) \times [1 + 1.5 + 3 + 5.5 + 4.5 + 2 + 1 + 2.5]
\]
\[
\int_R g(x, y) \, dA \approx 1 \times [1 + 1.5 + 3 + 5.5 + 4.5 + 2 + 1 + 2.5] = 21
\]

So the estimated value of the integral is **21**.

Would you like to know more details or have any further questions?

Here are five related questions:
1. How would the result change if the grid was refined?
2. What is the error associated with using the Midpoint Rule in this case?
3. How would this estimate change if you used the Trapezoidal Rule instead?
4. Can this method be applied to other types of functions or regions?
5. What would be the effect of increasing or decreasing the intervals for \(x\) and \(y\)?

**Tip:** The Midpoint Rule provides a more accurate approximation than the Trapezoidal Rule when the function is relatively smooth within the region of integration.

Learn how to estimate a double integral using the Midpoint Rule with a detailed example and solution. Explore the process of applying the Midpoint Rule to rectangular regions, along with step-by-step calculations and key values. Suitable for advanced college-level mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation