   Chapter 15.1, Problem 2E

Chapter
Section
Textbook Problem

If R = [0, 4] × [−1, 2], use a Riemann sum with m = 2, n = 3 to estimate the value of ∬ R ( 1 − x y 2 ) d A . Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.

(a)

To determine

To estimate: The value of given double integral for the lower right corners over the rectangular region R.

Explanation

Formula used:

The double integral of f over the rectangle R is,

Rf(x,y)dAlimm,ni=1mj=1nf(xi,yj)ΔA

Here, ΔA=lb , where l,b are the length and breadth of each rectangle.

The given function is f(x,y) .

The sample points of the lower right corner of each rectangle is denoted by (xi,yj) .

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

The Riemann sum constants are denoted by m, n.

Given:

The function, f(x,y)=(1xy2) .

The rectangular region, R=[0,4]×[1,2] .

The Riemann sum constants, m=2,n=3 .

Calculation:

Plot the given rectangle in the graph and pick the sample points at the lower right corners of each rectangle.

From Figure 1, it is observed that the lower right corners of each rectangle is (2,1),(2,0),(2,1),(4,1),(4,0),(4,1) and l=2,b=1 .

Therefore, ΔA=2 .

Substitute 2 for x and −1 for y in the given function f(x,y)=(1xy2) .

f(2,1)=(1(2(1)2))=(12)=1

Substitute 2 for x and 0 for y in the given function f(x,y)=(1xy2) .

f(2,0)=(1(2(0)2))=(10)=1

Substitute 2 for x and 1 for y in the given function f(x,y)=(1xy2) .

f(2,1)=(1(2(1)2))=(12)=1

Substitute 4 for x and −1 for y in the given function f(x,y)=(1xy2)

(b)

To determine

To estimate: The value of given double integral for the upper left corners over the rectangle R.

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