Chapter 14.6, Problem 33E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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+y2â‰¤9,â€‰0â‰¤zâ‰¤4}

Calculation:

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

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

âˆ­Qxyzdv=âˆ«âˆ’33âˆ«âˆ’9âˆ’x29âˆ’x2âˆ«04xyzâ€‰dzdydx

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

âˆ­Qxyzdv=âˆ«04âˆ«âˆ’33âˆ«âˆ’9âˆ’y29âˆ’y2xyzâ€‰dxdydzâ€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰=âˆ«âˆ’33âˆ«04âˆ«âˆ’9âˆ’y29âˆ’y2xyzâ€‰dxdzdyâ€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰=âˆ«04âˆ«âˆ’33âˆ«âˆ’9âˆ’x29âˆ’x2xyzâ€‰dydxdz=âˆ«âˆ’33âˆ«04âˆ«âˆ’9âˆ’x29âˆ’x2xyzâ€‰dydzdxâ€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰È€

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