(837, #7) Find the Riemann sum for f(x,y) = x2 - y2 on R = [0,5] x [0,2], with respect to the partition formed by the lines x = 1, 3, 4 and y = 1/2, 1

(837, #7) Find the Riemann sum for f(x,y) = x² - y² on R = [0,5] x [0,2], with respect to the partition formed by the lines x = 1, 3, 4 and y = 1/2, 1. Take (x_i^*, y_j^*) to be the upper, left corner of R_i,j .

Solution: The surface looks like a slide:

The partition given here is not standard, in that the rectangles do not have the same area:

The (x_i^*, y_j^*), the upper, left corners, are

(0,2)	(1,2)	(3,2)	(4,2)
(0,1)	(1,1)	(3,1)	(4,1)
(0, 1/2)	(1,1/2)	(3, 1/2)	(4,1/2)

The Riemann sum is

f(0,2) +	*f(1,2)2 +**	f(3,2) +	f(4,2) +
*f(0,1)(1/2) +**	f(1,1) +	*f(3,1)(1/2) +**	*f(4,1)(1/2) +**
*f(0,1/2)(1/2) +**	f(1,1/2) +	*f(3,1/2)(1/2) +**	*f(4,1/2)(1/2)**	= 247/8

by:rjm