   Chapter 15, Problem 1P

Chapter
Section
Textbook Problem

If [ x ] denotes the greatest integer in x, evaluate the integral ∬ R [ x + y ] d A where R = {(x, y) | 1 ≤ x ≤ 3, 2 ≤ y ≤ 5}.

To determine

To evaluate: The given integral.

Explanation

Given:

The function, f(x,y)=x+y.

The region R is R={(x,y)|1x3,2y5}.

Calculation:

Since the given region is the rectangular box in the first quadrant, separate the given region R into five parts, in which each region Ri has the points (x,y) provided i+2x+y<i+3. So, separate the integral into five parts as shown below.

Rx+ydA=i=15Rix+ydA=i=15(i+2)RidA=(3)R1dA+(4)R2dA+(5)R3dA+(6)R4dA+(7)R5dA=(3)A(R1)+(4)A(R2)+(5)A(R3)+(6)A(R4)+(7)A(R5)

From the condition of the sub region R1 and R5, it is observed that those regions are triangle with base and height one unit

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