Chapter 14.7, Problem 20E

### 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

# Volume In Exercises 15-20, use cylindrical coordinates to find the volume of the solid.Solid inside the sphere x 2 + y 2 + z 2 = 4 and above the upper nappe of the cone z 2 = x 2 + y 2

To determine

To calculate: Find out the Volume of Solid confined below the cone bysphere x2+y2+z2=4 and also find the volume above the upper nappe of the cone defined by the equation z2=x2+y2.

Explanation

Given:

Volume of Solid confined below the cone by sphere x2+y2+z2=4 and above its upper nappe

Upper nappe by the equation z2=x2+y2.

Formula used:

By triple integration of cylindrical coordinateâ€™s volume of solid can be defined as

V=âˆ­Qf(x,y,z)dV=âˆ­Qf(rcosÎ¸,rsinÎ¸,z)rdzdrdÎ¸

Rectangular conversion equations of cylindrical coordinates are,

x=rcosÎ¸y=rsinÎ¸z=z

Relation between x, y, and r is given as,

x2+y2=r2

As, sin2Î¸+cos2Î¸=1

Trigonometry formula,

sin3Î¸=3sinâˆ’1Î¸âˆ’4sin3Î¸

Calculation:

Deliberate the solid confined by the sphere by x2+y2+z2=4 and above the upper nappe of cone by z2=x2+y2.

Consider the available equation,

x2+y2+z2=4 â€¦â€¦ (1)

And,

z2=x2+y2 â€¦â€¦ (2)

Use equation (2) in equation (1) as,

x2+y2+z2=4z2+z2=42z2=4z=Â±2

Also,

x2+y2+z2=4r2+z2=4z2=4âˆ’r2z=4âˆ’r2

Use polar components in eq. (2)

x2+y2=z2r2=z2

Limits on z is,

râ‰¤zâ‰¤4âˆ’r2

Limits of r is,

0â‰¤râ‰¤2

Limits on Î¸ is,

0â‰¤Î¸â‰¤2Ï€

Therefore, volume of required solid is given as,

V=âˆ­Qf(x,y,z)dV=âˆ«02Ï€âˆ«02âˆ«r4âˆ’r2rdzdrdÎ¸=âˆ«02Ï€âˆ«02r[z]r4âˆ’r2

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