   Chapter 14.6, Problem 33E

Chapter
Section
Textbook Problem

Orders of Integration In Exercises 31-34, write a triple integral for f ( x , y , z ) = x y z over the solid region Q for each of the six possible orders of integration. Then evaluate one of the triple integrals. Q = { ( x , y , z ) :     x 2 + y 2 ≤ 9 ,     0 ≤ z ≤ 4 }

To determine

To Calculate: A triple integral for f(x,y,z)=xyz over the provided solid region Q where, Q={(x,y,z):x2+y29,0z4} for each of the six possible orders of integration and evaluating one of the triple integral.

Explanation

Given:

The provided solid region is Q={(x,y,z):x2+y29,0z4}

Calculation:

From provided Q region, z varies from 0 to 4, y varies from 9x2 to 9x2 and x varies from 3 to 3

Therefore, the required triple integral in order of dzdydx is given by,

Qxyzdv=339x29x204xyzdzdydx

Similarly, triple integral in other possible order is represented by,

Qxyzdv=04339y29y2xyzdxdydz=33049y29y2xyzdxdzdy=04339x29x2xyzdydxdz=33049x29x2xyzdydzdxȀ

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