Chapter 14.6, Problem 22E

### 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

# Volume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z = 9 − x 3 ,     y = − x 2 + 2 ,       y = 0 ,       z = 0 ,       x ≥ 0

To determine

To calculate: The volume of the solid bounded by with the help of triple integral if the graphs of equations is z=9x3,y=x2+2,y=0,z=0,x0

Explanation

Given: The given equation are z=9âˆ’x3,â€‰â€‰y=âˆ’x2+2,â€‰y=0,â€‰z=0,â€‰xâ‰¥0

Formula used: The Volume of the solid region Q can be calculated by V=âˆ­Qdxdydz

Calculation:

From the provided equations, it can be concluded that the limit of variable z varies from 0 to 9âˆ’x3, y varies from 0 to âˆ’x2+2 and x varies from 0 to 2. Thus the volume of the solid is given by, âˆ«02âˆ«0âˆ’x2+2âˆ«09âˆ’x3dzdydx=âˆ«02âˆ«0âˆ’x2+2[z]09âˆ’x3dydxâ€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰=âˆ«02âˆ«03âˆ’3x(9âˆ’x3)dydxâ€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰=âˆ«02[(9âˆ’x3)y]03âˆ’3xdxâ€‰â€‰â€‰â€‰â€‰

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