Final answer:
To compute the Riemann sum for the given double integral, divide the region of integration R into rectangles and evaluate the function at the sample points within each rectangle. The grid in the figure provides the dimensions of each rectangle. Use the formula for the Riemann sum to calculate the exact answer.
Step-by-step explanation:
To compute the Riemann sum for the given double integral, we need to divide the region of integration R into small rectangles and evaluate the function at the sample points within each rectangle. We are given the grid in the figure, so we can determine the dimensions of each rectangle. The interval [1,4] is divided into 3 equal subintervals, and the interval [2,6] is divided into 2 equal subintervals, resulting in a grid of 3x2 rectangles.
We can denote the sample points as (x_i, y_j), where x_i is the x-coordinate of the ith rectangle and y_j is the y-coordinate of the jth rectangle. We have S₃,₂, so the sample points are (2, 3), (3, 3), (4, 3), (2, 4), (3, 4), and (4, 4). For each rectangle, we evaluate the function 7x - 2y at the sample point, multiply it by the area of the rectangle, and sum up the results.
The area of each rectangle can be calculated as the difference between the x-coordinates and y-coordinates of adjacent sample points. In this case, the width of each rectangle is 1 (since adjacent x-coordinates differ by 1) and the height of each rectangle is 1 (since adjacent y-coordinates differ by 1).
Finally, we can write the Riemann sum as S₃,₂ = [(7*2 - 2*3)*1*1 + (7*3 - 2*3)*1*1 + (7*4 - 2*3)*1*1 + (7*2 - 2*4)*1*1 + (7*3 - 2*4)*1*1 + (7*4 - 2*4)*1*1]. We can simplify this expression to get the exact answer.