   Chapter 15.6, Problem 17E

Chapter
Section
Textbook Problem

Evaluate the triple integral.17. ∭ E x   d V , where E is bounded by the paraboloid x – 4y2 + 4z2 and the plane x = 4

To determine

To evaluate: The integral of ExdV .

Explanation

Given:

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

The region is E={(x,y,z)|0xπ,0yx+π,0zx} .

Calculation:

The given integral is, ExdV=D[4y2+4z24xdx]dA .

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

ExdV=D[x22]4y2+4z24dA=12D[(4)2(4y2+4z2)2]dA=12D[1616(y2+z2)2]dA=8D[1(y2+z2)2]dA

By applying polar coordinates, write the above differential equation will be,

ExdV=802π01[1(r2)2]

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