   Chapter 15.6, Problem 37E

Chapter
Section
Textbook Problem

Evaluate the triple integral using only geometric interpretation and symmetry.37. ∭ c   ( 4   +   5 x 2 y z 2 )   d V , where C is the cylindrical region x 2   +   y 2   ≤   4 ,   − 2   ≤   z   ≤ 2

To determine

To evaluate: The integral by using geometric interpretation and symmetry.

Explanation

Given:

The integral is C(4+5x2yz2)dV where C is the cylindrical region x2+y24 and 2z2 .

Calculation:

Write the given integral as C(4+5x2yz2)dV=C4dV+C5x2yz2dV (1)

But here the function f(x,y,z)=5x2yz2 is an odd function depending on y value a and C is also symmetrical about xz-plane.

Therefore, C5x2yz2dV=0 since ( abf(x)dx=0 if f(x) is an odd function)

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