Explanation Given: Calculation: Consider a torus generated by revolving a circle of radius ‘ a ’ around a vertical axis at a distance ‘ b ’ from its center b > a . Assume that the center of the circle is at x-axis and at a distance b from the y-axis. Then, the equation of the circle is ( x − b ) 2 + ( y − 0 ) 2 = a 2 ( x − b ) 2 = a 2 − y 2 x − b = ± a 2 − y 2 x = b ± a 2 − y 2 The given region is represented as follows: When the given circle is revolved about the y-axis, the volume generated will be calculated using Washer’s method. V = ∫ c d π [ { f ( y ) } 2 − { g ( y ) } 2 ] d y ” Outer radius will be, f ( y ) = b + a 2 − y 2 And, the inner radius will be, g ( y ) = b − a 2 − y 2 Interval of the solid will be [ − a , a ] . So, the volume generated will be, V = ∫ − a a π [ { b + a 2 − y 2 } 2 − { b − a 2 − y 2 } 2 ] d y = ∫ − a a π [ b 2 + ( a 2 − y 2 ) + 2 b a 2 − y 2 − b 2 − ( a 2 − y 2 ) + 2 b a 2 − y 2

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.2, Problem 56PS

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To verify: Kepler’s formula for the volume of a torus using Washers or Shells method.

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Chapter 6.2, Problem 55PS

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Chapter 6 Solutions

Calculus

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To verify: Kepler’s formula for the volume of a torus using Washers or Shells method.

Solution Summary: The author explains Kepler's formula for the volume of a torus using Washers or Shells method.

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