   Chapter 15.1, Problem 7E

Chapter
Section
Textbook Problem

A contour map is shown for a function f on the square R = [0, 4] × [0. 4].(a) Use the Midpoint Rule with m = n = 2 to estimate the value of ∬ R f ( x , y ) d A .(b) Estimate the average value of f. (a)

To determine

To estimate: The value of given double integral over the rectangle R by using midpoint rule.

Explanation

Formula used:

The double integral, Rf(x,y)dAi=1mj=1nf(x¯i,y¯j)ΔA ,

Where, ΔA=a2 , where a is the side of each square.

The given function is f(x,y) .

The mid points of each rectangle is denoted by (x¯i,y¯j) , where x¯i is the midpoint of [xi1,xi] and y¯i is the midpoint of [yj1,yj] .

The Riemann sum constants are denoted by m, n.

Given:

The rectangular region, R=[0,4]×[0,4] .

The Riemann sum constants, m=2,n=2 .

Calculation:

Plot the given rectangle in the graph and pick the mid points of each rectangle.

From Figure 1, it is observed that the mid points of each rectangle is (1,1),(1,3),(3,1)(3,3) and a=2

(b)

To determine

To estimate: The average value of given function.

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