   Chapter 15.6, Problem 10E

Chapter
Section
Textbook Problem

Evaluate the triple integral.10. ∭ E e z y ​   d V ,   where E   =   { ( x ,   y ,   z ) |   0   ≤   y   ≤   1 ,       y   ≤   x   ≤   1 ,     0   ≤   z     ≤   x y }

To determine

To evaluate: The given triple integral.

Explanation

Given:

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

The region is E={(x,y,z)|yx1,0y1,0zxy} .

Calculation:

The given integral is, EydV=01y10xyezydzdxdy .

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

EydV=01y1[ezy(1y)]0xydxdy=01y1[ezyy]0xydxdy=01y1[exyyye0yy]dxdy=01y1[exyy]dxdy

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

EydV=01[yexyx]y1dy=01[(ye1y(1))(yeyy(y))]dy=01[yeyyey+y2]dy=01[y(e1)yey+y2]dy

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

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