Chapter 15.7, Problem 27E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use cylindrical coordinates.

27. Find the mass and center of mass of the solid *S* hounded by the paraboloid *z* = 4*x*^{2} + 4*y ^{2}* and the plane

To determine

**To find:** The mass and center of the given solid.

Explanation

**Given:**

The solid *S* is bounded by the paraboloid

The density function is constant say *K*.

**Calculation:**

The mass of the given solid *S* is,
*z* varies from
*a.* To find the limits of *r* solve the given equations.

Therefore, the mass of the given solid is given by,

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

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

Integrate with respect to

It is given that the density is constant and
*K* is the constant.

By the symmetric of the given function, it is observed that *x* and *y* coordinates of the center is 0. Thus, the third coordinate is given by,

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

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