   Chapter 15.6, Problem 14E

Chapter
Section
Textbook Problem

Evaluate the triple integral.14. ∭ E ( x   −   y )   d V , where E lies enclosed by the surface z = x2 – 1, z = 1 – x2, y = 0 and y = 2

To determine

To evaluate: The given triple integral.

Explanation

Given:

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

The region E is enclosed by the surfaces z=x21,z=1x2,y=0,y=2 ..

Calculation:

Solve the equations of z to find the limits of x.

x21=1x22x22=02(x21)=0x21=0

Solving this will yield x=±1 . Hence, E={(x,y,z)|1x1,0y2,x21z1x2} .

Thus, the given integral is, E(xy)dV=1102x211x2(xy)dzdydx .

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

E(xy)dV=1102(xy)[z]x211x2dydx=1102(xy)[1x2x2+1]dydx=1102(xy)[22x2]dydx=1102(2x2x32y+2x2y)dydx

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

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