Chapter 15.6, Problem 41E

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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*.

**41**. *E.* is the cube given by 0 ≤ *x* ≤ *a,* 0 ≤* y* ≤* a,* 0 ≤ z ≤ *a*; *ρ*(*x, y, z*) = *x*^{2}* + y*^{2}* + z*^{2}

To determine

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

Explanation

**Given:**

*E* is cube given by

**Calculation:**

Mass of the solid,

The mass of the solid is,

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

From the given observe that *x* varies from 0 to *a*, *y* varies from 0 to *a* and *z* varies from 0 to *a*. Here the density is given by

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

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

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

Thus, the mass of the solid is

To find 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 *y* and apply the limit of it.

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

Substitute the value in equation (1),

Center of mass located at

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

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