Chapter 15.6, Problem 39E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Find the mass and center of mass of the solid *E* with the given density function *ρ*.

**39**. *E* lies above the *xy*-plane and below the paraboloid *z =* 1 – *x*^{2} – *y*^{2}; *ρ* (*x, y _{,} z*) = 3

To determine

**To find:** The mass and center of mass of the solid *E* with density

Explanation

**Given:**

*E* lies above the *xy*-plane and below the paraboloid

**Calculation:**

Mass of the solid ,

The mass of the solidis,

Here, the density function is given by
*E* is the region that is occupied by the solid.

Covert the boundary of *z* into polar form. Substitute
*z* varies from 0 to

The value of *r* varies from 0 to 1 and

Integrate the above integral with respect to *r* and apply the limit of it.

Integrate the above integral with respect to

Thus, the mass of the solid is

For finding the center of mass,

Center of mass located at

Integrate the numerator part with respect to *z* and apply the limit of it.

Integrate the above integral with respect to *r* and apply the limit of it.

Integrate the above integral with respect to

Substitute the value in equation (1),

Center of mass located at

Integrate the numerator part with respect to *z* and apply the limit of it

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