Explanation Given information: Let T be the torus obtained by rotating the circle in the y z -plane of the radius a centered at ( 0 , b , 0 ) about the z − a x i s .(Figure 24 ).We assume that b > a > 0 . Calculation: Consider the equation of the circle of radius a as given below; ( y − b ) 2 + z 2 = a 2 Now consider the rotation around y − a x i s instead of z − a x i s . Let us again consider; z = a 2 − ( y − b ) 2 d z d y = 1 2 − 2 ( y − b ) 2 a 2 − ( y − b ) 2 d z d y = − ( y − b ) a 2 − y 2 + 2 b y − b 2 The value of elementary arc length is, d s = d y 2 + d z 2 = 1 + ( d z d y ) 2 . d y = 1 + ( − ( y − b ) a 2 − y 2 + 2 b y − b 2 ) 2 . d y = ( a 2 − ( y − b ) 2 + ( y − b ) 2 a 2 − y 2 + 2 b y − b 2 ) To determine (b) To show: The area of a torus is A r e a ( T ) = 4 π 2 a b .

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 17.4, Problem 45E

To determine

(a)

To show:The area of a torus is given by Area(T)=4π∫b−ab+aay a 2 − ( b−y ) 2 dy.

To determine

(b)

To show:

The area of a torus is Area(T)=4π2ab.

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Chapter 17.4, Problem 44E

Chapter 17.4, Problem 46E

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Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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(a) To show: The area of a torus is given by A r e a ( T ) = 4 π ∫ b − a b + a a y a 2 − ( b − y ) 2 d y .

Solution Summary: The author explains how the area of a torus is given by Area(T)=4pi displaystyle

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(a)

To show:The area of a torus is given by Area(T)=4π∫b−ab+aay a 2 − ( b−y ) 2 dy.

(b)

To show:

The area of a torus is Area(T)=4π2ab.

Want to see the full answer?

Chapter 17 Solutions

(a) To show: The area of a torus is given by A r e a ( T ) = 4 π ∫ b − a b + a a y a 2 − ( b − y ) 2 d y .

Solution Summary: The author explains how the area of a torus is given by Area(T)=4pi displaystyle

Videos

(a) To show:The area of a torus is given by Area(T)=4π∫b−ab+aay a 2 − ( b−y ) 2 dy.

(b) To show: The area of a torus is Area(T)=4π2ab.

Want to see the full answer?

Chapter 17 Solutions

(a)

To show:The area of a torus is given by Area(T)=4π∫b−ab+aay a 2 − ( b−y ) 2 dy.

(b)

To show:

The area of a torus is Area(T)=4π2ab.