   Chapter 14.7, Problem 23E

Chapter
Section
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=dVV=rdzdrdθ

Rectangular conversion equations of cylindrical coordinates:

x=rcosθy=rsinθz=z

Calculation:

Consider the provided figure:

From the cone figure:

Bounds on z are

0zh(1rr0)

Here, h and r0 are constant

Bounds on r are

0rr0

Bounds on θ are

0θ2π

Volume of the cone is

V=dV=02π0r00h(1rr0)rdzdrdθ=02π0r0[rz]0h(1rr0)drdθ=02π0r0rh(1rr0)drdθ

On further integration

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