Consider the solid that is bounded above by x² + y² + z? = 16 and below by z = Jx² + y². Set up only the appropriate triple integrals in rectangular, cylindrical, and spherical coordinates needed to find the volume of the solid.

Consider the solid that is bounded above by x² + y² + z? = 16 and below by z = Jx² + y². Set up only the appropriate triple integrals in rectangular, cylindrical, and spherical coordinates needed to find the volume of the solid.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter9: Surfaces And Solids

Section9.CR: Review Exercises

Problem 22CR

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Question

Consider the solid that is bounded above by x² + y² + z? = 16 and below by z = Jx² + y². Set up only
the appropriate triple integrals in rectangular, cylindrical, and spherical coordinates needed to find the
volume of the solid.

Expert Solution

Step 1

Given:

The solid that is bounded above by x²+ y²+ z²=16 and below by

Step 2

To set up the triple integral in rectangular coordinate first solve the equation x²+y²+z²=16 for z, we get

The solid is bounded above the upper part of the sphere

Therefore, the volume of the solid in rectangular coordinate is

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