   Chapter 15.1, Problem 3E

Chapter
Section
Textbook Problem

(a) Use a Riemann sum with m = n = 2 to estimate the value of ∬ R x e − x y d A , where R = [0, 2] × [0, l]. Take the sample points to be upper right corners.(b) Use the Midpoint Rule to estimate the integral in part (a).

(a)

To determine

To estimate: The value of given double integral by taking the sample points from upper 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 is f(x,y)=xexy .

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

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

Calculation:

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

From Figure 1, it is observed that the upper right corners of each rectangle is (1,12),(1,1),(2,12),(2,1) and l=1,b=12 .

Therefore, ΔA=12 .

Substitute x=1 and y=12 in the given function f(x,y)=xexy .

f(1,12)=1×e(1)12=e12

Substitute x=1 and y=1 in the given function f(x,y)=xexy

(b)

To determine

To estimate: The value of given double integral over the rectangle R.

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