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

Transcript text: A contour map is shown for a function $f$ on the square $R=[0,4] \times[0,4]$ Use the Midpoint Rule with $m=n=4$ to estimate the value for $\iint_{R} f(x, y) d A$

Solution

Solution Steps

Step 1: Divide the Region into Subrectangles

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 $.

Step 2: Identify the Midpoints of Each Subrectangle

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)

Step 3: Evaluate the Function at Each Midpoint

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 $

Step 4: Apply the Midpoint Rule

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

\[ \iint_R f(x, y) \, dA \approx 1 \cdot 1 \cdot 237 = 237 \]

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