Chapter 15.6, Problem 40E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

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

**40**. *E* is bounded by the parabolic cylinder *z* = 1 *– y ^{2}* and the planes

To determine

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

Explanation

**Given:**

*E* is bounded by the parabolic cylinder

**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.

From the given observe that, *y* varies from
*z* varies from 0 to
*x* varies from 0 to

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

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

On further simplification find the value of *m*.

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

On further simplification find the value of *m*.

Thus, the mass of the solid is

To calculate the center of mass,

Center of mass located at

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

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

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

Simplify further.

Substitute the value in equation (1),

Center of mass located at

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