   Chapter 15, Problem 1RE

Chapter
Section
Textbook Problem

A contour map is shown for a function f on the square R = [0, 3] × [0, 31. Use a Riemann sum with nine terms to estimate the value of ∬ R f ( x , y )   d A . Take the sample points to be the upper right comers of the squares. To determine

To estimate: The value of Rf(x,y)dA by taking the sample points in the upper right corner of each square.

Explanation

Formula used:

The double integral, Rf(x,y)dAlimm,ni=1mj=1nf(xi,yj)ΔA ,

Here, ΔA=a2 where a is the length of the side of the square.

The sample points of the upper right corner of each square is denoted as (xi,yj) .

The image value of the sample points under the function f is denoted as f(xi,yj) .

And, m, n is the Riemann sum constants.

Given:

The rectangle, R=[0,3]×[0,3] .

The Riemann sum constants, m=3,n=3 .

Calculation:

Plot the given rectangle in the graph and pick the sample points at the upper right corners of each square as shown below in Figure 1.

From Figure 1, it is observed that the upper right corners of each square is (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3) and a=1 .

Therefore, ΔA=1 . Again, from Figure 1, it is observed that f(1,1)=2.7 , f(1,2)=4

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