Explanation After determining radii of the washers, obtain the volume of the torus to prove that it is equal to 2 π 2 a b 2 . Given: The torus (doughnut-shaped solid) in Figure 15 is obtained by rotating the circle ( x − a ) 2 + y 2 = b 2 around the y -axis, assuming that a > b . Calculation: The torus (doughnut-shaped solid) in Figure 15 is obtained by rotating the circle ( x − a ) 2 + y 2 = b 2 around the y -axis, assuming that a > b . The cross-sections are washers with outer radius R = a + ( b 2 − y 2 ) and inner radius r = a − ( b 2 − y 2 )

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.3, Problem 56E

To determine

That the volume of the torus is 2π2ab2 . Then, evaluate the integral by interpreting it as the area of a circle.

Solution:

The volume of the torus thus obtained is 2π2ab2 .

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Chapter 6.3, Problem 55E

Chapter 6.3, Problem 57E

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Chapter 6 Solutions

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That the volume of the torus is 2 π 2 a b 2 . Then, evaluate the integral by interpreting it as the area of a circle. Solution: The volume of the torus thus obtained is 2 π 2 a b 2 .

Solution Summary: The author determines that the volume of the torus is 2pi 2ab2. The integral is one-half the area of a circle.

Concept explainers

Videos

That the volume of the torus is 2π2ab2 . Then, evaluate the integral by interpreting it as the area of a circle.

Solution:

The volume of the torus thus obtained is 2π2ab2 .

Want to see the full answer?

Chapter 6 Solutions

That the volume of the torus is 2 π 2 a b 2 . Then, evaluate the integral by interpreting it as the area of a circle. Solution: The volume of the torus thus obtained is 2 π 2 a b 2 .

Solution Summary: The author determines that the volume of the torus is 2pi 2ab2. The integral is one-half the area of a circle.

Concept explainers

Videos

That the volume of the torus is 2π2ab2 . Then, evaluate the integral by interpreting it as the area of a circle. Solution: The volume of the torus thus obtained is 2π2ab2 .

Want to see the full answer?

Chapter 6 Solutions

That the volume of the torus is 2π2ab2 . Then, evaluate the integral by interpreting it as the area of a circle.

Solution:

The volume of the torus thus obtained is 2π2ab2 .