(837, #3) Calculate the Riemann sum for f(x,y) = x2 +
4 y on R = [0,2] x [0,3] for a partition based on the lines x = 1 and y
= 1, y =2. Take (xi*, yj*)
to be the upper rightcorner of each rectangle of the partition.
Solution: This is the solid whose volume we wish to approximate:
The approximating rectangular blocks, whose height is the value of
f(x,y) at the upper right hand corner at each block of the 2 x 3 partition
of the domain, are pictured below.
The (xi*, yj*) values for the partition are (1,1), (1,2), (1,3) and (2,1), (2,2), (3,3).
Thus the Riemann sum, the sum of the volume of the blocks is
( f(1,1) + f(1,2) + f(1,3) + f(2,1) + f(2,2) + f(3,3)) ((2
- 0)/2)) ((3 - 0)/3) = 63.
Students using Maple can avoid the labor of computation by using
the command
> sum(sum(f (i,j), i = 1..2),j = 1..3);
63
by:rjm