Chapter 14.7, Problem 23E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Using Cylindrical Coordinates In Exercises 23-28, use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure.Find the volume of the cone.

To determine

To calculate: The volume of Cone the equation of the cone is z=h(1rr0)

Explanation

Given:

The provided figure of cone is:

Formula Used:

Volume of cone by triple integration is:

V=âˆ­dVV=âˆ­rdzdrdÎ¸

Rectangular conversion equations of cylindrical coordinates:

x=rcosÎ¸y=rsinÎ¸z=z

Calculation:

Consider the provided figure:

From the cone figure:

Bounds on z are

0â‰¤zâ‰¤h(1âˆ’rr0)

Here, h and r0 are constant

Bounds on r are

0â‰¤râ‰¤r0

Bounds on Î¸ are

0â‰¤Î¸â‰¤2Ï€

Volume of the cone is

V=âˆ­dV=âˆ«02Ï€âˆ«0r0âˆ«0h(1âˆ’rr0)rdzdrdÎ¸=âˆ«02Ï€âˆ«0r0[rz]0h(1âˆ’rr0)drdÎ¸=âˆ«02Ï€âˆ«0r0rh(1âˆ’rr0)drdÎ¸

On further integration

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