Explanation Given: The inequality: ( x 2 + y 2 + z 2 + 8 ) 2 ≤ 36 ( x 2 + y 2 ) Formula used: The volume of the torus is: V = ∫ 0 2 π ∫ r 1 r 2 ∫ z 1 z 2 r d z d r d θ The integration formula: ∫ a 2 − x 2 d x = x ( a 2 − x 2 ) 2 + a 2 2 sin − 1 ( x a ) + C Calculation: Consider the provided inequality, ( x 2 + y 2 + z 2 + 8 ) 2 ≤ 36 ( x 2 + y 2 ) Convert the inequality for the points ( x , y , z ) in the cylindrical form that is for the point ( r cos θ , r sin θ , z ) From this, ( ( r cos θ ) 2 + ( r cos θ ) 2 + z 2 + 8 ) 2 ≤ 36 ( ( r cos θ ) 2 + ( r sin θ ) 2 ) ( r 2 + z 2 + 8 ) 2 ≤ 36 r 2 Take square root both the sides to get, r 2 + z 2 + 8 ≤ 6 r Form this, r 2 − 6 r + z 2 + 8 ≤ 0 r 2 − 2 ( 3 ) r + 9 − 9 + z 2 + 8 ≤ 0 ( r − 3 ) 2 + z 2 ≤ 1 …… (1) Draw the above equation in the r-z plane. The graph obtained is shown below. Thus, the equation (1) represents a torus about z -axis. Therefore, the volume of the torus is: V = ∫ 0 2 π ∫ r 1 r 2 ∫ z 1 z 2 r d z d r d θ The limits are as follows. − 1 − ( r − 3 ) 2 ≤ z ≤ 1 − ( r − 3 ) 2 2 ≤ r ≤ 4 0 ≤ θ ≤ 2 π Put the limits in the formula to get, V = ∫ 0 2 π ∫ 2 4 ∫ − 1 − ( r − 3 ) 2 1 − ( r − 3 ) 2 r d z d r d θ Solve the iterated integral to get the volume of the torus, V = 2 ∫ 0 2 π ∫ 2 4 ∫ 0 1 − ( r − 3 ) 2 r d z d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( ∫ 0 1 − ( r − 3 ) 2 r d z ) d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( r 1 − ( r − 3 ) 2 ) d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( r − r 2 + 6 r − 9 ) d r d θ On further simplification, V = 2 ∫ 0 2 π ∫ 2 4 ( ( − 1 2 ( − 2 r + 6 ) + 3 ) − r 2 + 6 r − 8 ) d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( − 1 2 ( − 2 r + 6 ) − r 2 + 6 r − 8 + 3 − r 2 + 6 r −

11th Edition

ISBN: 9781337275576

Author: Larson, Ron, Edwards, Bruce

Publisher: Cengage Learning,

See similar textbooks

Videos

Question

Chapter 14.7, Problem 47E

To determine

To calculate: The volume of the region of points (x,y,z) such that (x2+y2+z2+8)2≤36(x2+y2).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.7, Problem 46E

Chapter 14.8, Problem 1E

Students have asked these similar questions

SOLVE WITHOUT ARTIFICIAL INTELLIGENCE BY HAND,LEGIBLE PLEASE The region in the first octant bounded by the coordinates planes and the planes x + z = 1, y+2z = 2. In order of dzdxdy

Second question of question (d) please as soon as possible

Help me please

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 region of points ( x , y , z ) such that ( x 2 + y 2 + z 2 + 8 ) 2 ≤ 36 ( x 2 + y 2 ) .

Solution Summary: The author explains how to calculate the volume of the region of points (x,y,z).

Videos

To calculate: The volume of the region of points (x,y,z) such that (x2+y2+z2+8)2≤36(x2+y2).

Want to see the full answer?

Chapter 14 Solutions