   Chapter 15.6, Problem 13E

Chapter
Section
Textbook Problem

Evaluate the triple integral.13. ∭ E 6 x y   d V , where E lies under the plane z = 1 + x + y and above the region in the .xy-plane bounded by the curves y = x ,y = 0. and x = 1

To determine

To evaluate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=6xy .

The region E lies under the plane z=1+x+y and above the region bounded by the curves y=x,y=0,x=1 .

Calculation:

From the given conditions, it is observed that, E={(x,y,z)|0x1,0yx+π,0z1+x+y} . Then, the given integral becomes, E6xydV=010x01+x+y6xydzdydx .

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

E6xydV=010x6xy[z]01+x+ydydx=010x6xy[1+x+y0]dydx=010x6xy(1+x+y)dydx=6010x(xy+x2y+xy2)dydx

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

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