   Chapter 15.2, Problem 69E

Chapter
Section
Textbook Problem

Use geometry or symmetry, or both, to evaluate the double integral.69. ∬ D ( a x 3 + b y 3 + a 2 − x 2 )   d A , D =   [ − a ,   a ]   × [ − b ,   b ]

To determine

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

Explanation

Definition used:

Odd function: If f is a function and f(x)=f(x) , then f is said to be an odd function.

Formula used:

If f is an odd function, then aaf(x)dx=0 . (1)

Given:

The function is, f(x,y)=ax3+by3+a2x2 .

The domain D is, D=[a,a]×[b,b] .

Calculation:

From the given conditions,

Df(x,y)dA=D(ax3+by3+a2x2)dA=Dax3dA+Dby3dA+Da2x2dA

Here, first and second integrand are odd functions with respect to x and y, respectively, because, Dax3d

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