   Chapter 15.6, Problem 38E

Chapter
Section
Textbook Problem

Evaluate the triple integral using only geometric interpretation and symmetry.38. ∭ B   ( z 3   +   sin   y   +   3 )   d V , where B is the unit ball x2 + y2 + z2 ≤ 1

To determine

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

Explanation

Given:

The integral is B(z3+siny+3)dV where B is the unit ball x2+y2+z21 .

Calculation:

Write the given integral as,

B(z3+siny+3)dV=Bz3dV+BsinydV+B3dV (1)

But here the function f(x,y,z)=z3 is an odd function depending on z value and B is also symmetrical about xy-plane and the function f(x,y,z)=siny is an odd function depending on y value and B is also symmetrical about xz-plane.

Therefore, Bz3dV=0 and BsinydV=0 since ( abf(x)dx=0 if f(x) is an odd function)

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