Questions: A contour map is shown for a function f on the square R=[0,4] × [0,4] Use the Midpoint Rule with m=n=4 to estimate the value for ∫∫R f(x, y) d A

Solution Steps

The region $R = [0, 4] \times [0, 4]$ is divided into $m = n = 4$ subrectangles. Each subrectangle will have dimensions $\Delta x = \Delta y = 1$.

The midpoints of each subrectangle are:

• (0.5, 0.5), (1.5, 0.5), (2.5, 0.5), (3.5, 0.5)
• (0.5, 1.5), (1.5, 1.5), (2.5, 1.5), (3.5, 1.5)
• (0.5, 2.5), (1.5, 2.5), (2.5, 2.5), (3.5, 2.5)
• (0.5, 3.5), (1.5, 3.5), (2.5, 3.5), (3.5, 3.5)

Using the contour map, approximate the function values at the midpoints:

• $f(0.5, 0.5) \approx 23$
• $f(1.5, 0.5) \approx 15$
• $f(2.5, 0.5) \approx 15$
• $f(3.5, 0.5) \approx 23$
• $f(0.5, 1.5) \approx 15$
• $f(1.5, 1.5) \approx 5$
• $f(2.5, 1.5) \approx 5$
• $f(3.5, 1.5) \approx 15$
• $f(0.5, 2.5) \approx 15$
• $f(1.5, 2.5) \approx 5$
• $f(2.5, 2.5) \approx 5$
• $f(3.5, 2.5) \approx 15$
• $f(0.5, 3.5) \approx 23$
• $f(1.5, 3.5) \approx 15$
• $f(2.5, 3.5) \approx 15$
• $f(3.5, 3.5) \approx 23$

The Midpoint Rule for double integrals is given by: $$\iint_R f(x, y) \, dA \approx \Delta x \Delta y \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_i, y_j)$$ Here, $\Delta x = \Delta y = 1$, and the sum of the function values at the midpoints is: $$23 + 15 + 15 + 23 + 15 + 5 + 5 + 15 + 15 + 5 + 5 + 15 + 23 + 15 + 15 + 23 = 237$$

Final Answer