Final answer:
Approximating the double integral of f(x, y) = x + y over the region R involves using a Riemann sum to add the products of function values at specified points and the areas of subrectangles that are entirely within the semicircle and above the x-axis.
Step-by-step explanation:
The task involves approximating the double integral of f(x, y) = x + y over the region R. The region R is bounded above by the semicircle y = √(25-x²) and below by the x-axis. The integrand represents a simple plane in the XY-plane, increasing linearly with x and y. For approximation, the given partitions divide the region into a set of subrectangles, and the function values at the lower left corner of each subrectangle that lies within the region R are considered for the Riemann sum approximation of the integral.
The partition splits the x-domain into segments at x = -5, -2.5, 0, 1.25, 2.5, 5, and the y-domain into segments at y = 0, 2.5, 5. To approximate the double integral, we calculate the value of the function at each (x_k, y_k) within the semicircle bounds and then sum the products of these function values with the area of the corresponding subrectangles. Since the region R is a semicircle, not all rectangles will be fully within the bounds; we only include those that lie entirely within the semicircle.
The approximation process involves summing the products of the function values and the areas of the relevant subrectangles, ensuring that we stay within the boundary defined by the semicircle and the x-axis. The overall result will be an estimated value of the double integral over the region.