(837, #11) Estimate

where R = [0,1] x [0,1]. Use the midpoint rule and the following number of squares:

1, 4 16, 64, 256, 1024.

Solution: Let N be a nonnegative integer. Partion R into squares by means of the points

for i = 0..2^N and j = 0..2^N. Note that the corresponding partition has 2^N 2^N = 2^2N rectangles. Furthermore If R_{i, j} is the rectangle whose lower left and upper right

corners are, respectively,

then

is the midpoint of R_{i, j} .

Thus, the Maple code for eestimating the double integral here is

> f := (x,y)-> exp(-x^2-y^2):

S := N -> sum(sum(f( (i-0.5)/2^N, (j-0.5)/2^N), i = 1..2^N), j = 1..2^N)*(1/4^N);

Evaluating S(N) for N = 0..5 we get the following tabulation.

A graph of the underlying solid shows that the estimate is reasonable: The average value of the function f(x,y) is about 1/2; the domain has area 1. The estimates should be in the neighborhood of 1/2, which they are.

by:rjm