(a) Explanation Given: The solid region is given as: Formula used: The volume of the solid in cylindrical coordinates is calculated using the formula: V = ∫ θ 1 θ 2 ∫ r 1 r 2 ∫ z 1 z 2 r d z d r d θ The rectangular conversion equations for the cylindrical coordinates are given as: x = r cos θ y = r sin θ z = z Calculation: The limits of r , θ and z for the bounded solid is as follows: The limits for z are calculated as: 2 ≤ z ≤ 8 − x 2 − y 2 2 ≤ z ≤ 8 − r 2 The limits for r are calculated as: 0 ≤ r ≤ 2 The limits for θ are calculated as: 0 ≤ θ ≤ 2 π Put the limits of r , θ and z in the formula for the volume, V = ∫ 0 2 π ∫ 0 2 ∫ 2 8 − r 2 r d z d r d θ On solving the above integral, the volume of the bounded solid is calculated to be: V = ∫ 0 2 π ∫ 0 2 ∫ 2 8 − r 2 r d z d r d θ = ∫ 0 2 π ∫ 0 2 ( ∫ 2 8 − r 2 r d z ) d r d θ = ∫ 0 2 π ∫ 0 2 ( r [ z ] 2 8 − r 2 ) d r d θ = ∫ 0 2 π ∫ 0 2 r ( 8 − r 2 − 2 ) d r d θ Simplifying further, V = ∫ 0 2 π ∫ 0 2 r 8 − r 2 d r d θ − ∫ 0 2 π ∫ 0 2 2 r d r d θ = ∫ 0 2 π ( (b) To determine To calculate: The volume of the solid bounded below by the plane z = 2 and above the sphere x 2 + y 2 + z 2 = 8 using spherical coordinates.

11th Edition

ISBN: 9781337275576

Author: Larson, Ron, Edwards, Bruce

Publisher: Cengage Learning,

See similar textbooks

Videos

Question

Chapter 14, Problem 18PS

(a)

To determine

To calculate: The volume of the solid bound below by the plane z=2 and above the sphere x2+y2+z2=8 using cylindrical coordinates.

(b)

To determine

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using spherical coordinates.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14, Problem 16PS

Chapter 15.1, Problem 1E

Students have asked these similar questions

Use polar coordinates to find the volume the volume of the solid above the cone z = (x2 + y2)1/2 and below the sphere x2 + y2 +z2 = 25

Use cylindrical coordinates to find the volume of the solid. Solid inside x² + y² + z² = 16 and outside z = 2 + y²

Use cylindrical coordinates to find the volume of the solid

Chapter 14 Solutions

Calculus

Ch. 14.1 - CONCEPT CHECK Iterated Integral Explain what is...Ch. 14.1 - Prob. 2E Ch. 14.1 - Question: Evaluate the integral: 0x(2xy)dy Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The volume of the solid bound below by the plane z = 2 and above the sphere x 2 + y 2 + z 2 = 8 using cylindrical coordinates.

Videos

To calculate: The volume of the solid bound below by the plane z=2 and above the sphere x2+y2+z2=8 using cylindrical coordinates.

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using spherical coordinates.

Want to see the full answer?

Chapter 14 Solutions