   Chapter 15.2, Problem 67E

Chapter
Section
Textbook Problem

Use geometry or symmetry, or both, to evaluate the double integral.67. ∬ D ( 2 x + 3 y )   d A ,D is the rectangle 0 ≤ x ≤ a, 0 ≤ y ≤ b

To determine

To calculate: The value of given double integral over D using symmetry or geometry or both.

Explanation

Given:

The function is, f(x,y)=2x+3y .

The domain D is the rectangle 0xa,0yb .

Calculation:

From the given conditions,

Df(x,y)dA=D(2x+3y)dA=0a0b(2x+3y)dA=0a0b2xdA+0a0b3ydA

The first integrand 0a0b2xdA represents the volume of the solid lies below the plane z=2x and above the given rectangle D.

So the region obtained is a triangular cylinder with the length b. Note that each of the cross section has the base a and the height 2a

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