Chapter 15.7, Problem 25E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use cylindrical coordinates.

**25**. (a) Find the volume of the region *E* that lies between the paraboloid z = 24 - *x ^{2} - y^{2}* and the cone

(b) Find the centroid of *E* (the center of mass in the case where the density is constant).

(a)

To determine

**To find:** The volume of the given solid by using cylindrical coordinates.

Explanation

**Given:**

The region *E* lies between the paraboloid

**Formula used:**

If
*E* given by

The cylindrical coordinates

**Calculation:**

From the given equations, the value of *z* is obtained below.

The above equation is a quadratic equation.

Thus, solve the equation and obtain the value of

Therefore, *r* varies from 0 to 4, and
*z* varies from 2*r* to

Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the volume of the given solid is computed as follows.

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

(b)

To determine

**To find:** The centroid of the given region *E*.

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