   Chapter 15.1, Problem 31E

Chapter
Section
Textbook Problem

Calculate the double integral.31. ∬ R x sin ( x + y ) d A ,   R = [ 0 , π / 6 ] × [ 0 , π / 3 ]

To determine

To calculate: The value of given double integral over the region R.

Explanation

Given

The rectangular region is, R=[0,π6]×[0,π3] .

Calculation:

The value of the given double integral is,

Rxsin(x+y)dA=0π6[0π3xsin(x+y)dy]dx=0π6[xcos(x+y)]0π3dx=0π6[xcosxxcos(x+π3)]dx=0π6xcosxdx0π6xcos(x+π3)dx

Use integration by parts to each of the integrands,

Rxsin(x+y)dA=[xsinx+cosx]0π6[xsin(x+π3)+cos(x+π3)]0π6

Apply the limit values,

Rxsin(x+y)dA={[(

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