(837, #5)Calculate the Riemann sum for f(x,y) = x y - y² over R = [0, 5] x [0,4] with repect to the partition formed by the lines x = 1, 2, 3, 4 and y = 2. Take (x_i^*, y_j^*) to be the center of R_{i, j} .

Solution: The surface of this problem has both positive and negative values, so the Riemann sums cannot be interpreted as aapprox imating volumes. They are, rather, the algebraic difference of what is above - what is below the xy-plane.

The blocks of the Riemann sum are displayed below:

The centers of each subrectangle are at (1/2, 1), (3/2, 1), (5/2, 1), (7/2, 1), (9/2,1),
(1/2, 2), (3/2, 2), (5/2, 2), (7/2, 2), (9/2,2) and each subretangle has area 2, so the

Riemann sum for the given partition is

The Maple code for computing this sum is

> sum(sum(f(i-1/2, 2*j-1), i= 1..5),j = 1..2)*2;

The actual value of the corresponding double integral is -20/3, so the approximation is not very accurate.

by:rjm