   Chapter 15.1, Problem 50E

Chapter
Section
Textbook Problem

Use symmetry to evaluate the double integral.50. ∬ R ( 1 + x 2 sin y + y 2 sin x ) d A ,   R = { ( π , π ) × [ − π , π ] }

To determine

To find: The value of given double integral over R .

Explanation

Given:

The function is f(x,y)=1+x2siny+y2sinx .

The rectangular region, R=[π,π]×[π,π] .

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 g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (1)

If f is an odd function, then aaf(x)dx=0 (2)

Calculation:

By (1), the value of the given double integral is,

Rf(x,y)dA=ππππ(1+x2siny+y2sinx)dydx=ππππ(1)dydx+ππππx2sinydydx+

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