   Chapter 15.6, Problem 12E

Chapter
Section
Textbook Problem

Evaluate the triple integral.12. ∭ E sin y   d V , where E lies below the plane z = x and above the triangular region with vertices (0, 0, 0). ( π , 0, 0). and (0, π , 0)

To determine

To evaluate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=siny .

The region E is the triangle with vertices (0,0,0),(π,0,0),(0,π,0) .

Calculation:

From the given conditions of the region E, it is observed that the equations of the sides of the triangle are πx and x. Hence, E={(x,y,z)|0xπ,0yx+π,0zx} . Thus, the given integral is, EsinydV=0π0πx0xsinydzdydx .

Integrate the given integral with respect to z and apply the limit of it.

EsinydV=0π0πxsiny[z]0xdydx=0π0πxsiny[x0]dydx=0π0πxxsinydydx

Integrate the given integral with respect to y and apply the limit of it.

EsinydV=0πx[cosy]0πxdx=0πx[cos(πx)cos(0)]dx=0π[xcos(πx)+x.1]dx=0πxcos(πx)dx+0πxdx

On further simplification, the value of the integral becomes,

EsinydV=0πxcos(x+π)dx+0πxdx=0πxcos(x+π)dx+[x22]0π=0πxcos(x+π)dx+[(π)2(0)22]=0πxcos(x+π)dx+π22

Simplify further

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