   Chapter 15.6, Problem 9E

Chapter
Section
Textbook Problem

Evaluate the triple integral.9. ∭ E y ​   d V ,   where E   =   { ( x ,   y ,   z ) |   0   ≤   x   ≤   3 ,       0   ≤   y   ≤   x ,     x   −   y   ≤   z     ≤   x   +   y }

To determine

To evaluate: The integral of EydV.

Explanation

Given:

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

The region is E={(x,y,z)|0x3,0yx,xyzx+y}.

Calculation:

The given integral is, EydV=030xxyx+yydzdydx.

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

EydV=030xy[z]xyx+ydydx=030xy[(x+y)(xy)]dydx=030xy[2y]dydx=030xy2dydx

Integrate the given integral with respect to y and apply the limit

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