Explanation Given: Solid torus with radii r and R . The equation of the circle is ( x − R ) 2 + y 2 = r 2 Show the theorem 7 (Integrals of symmetric Functions): Apply the condition as follows if the function f is continuous on [– a , a ]. a) If f is even [ f ( − x ) = f ( x ) ] , then ∫ − a a f ( x ) d x = 2 ∫ 0 a f ( x ) d x . b) If f is odd [ f ( − x ) = − f ( x ) ] , then ∫ − a a f ( x ) d x = 0 . Calculation: Show the equation as below: ( x − R ) 2 + y 2 = r 2 y 2 = r 2 − ( x − R ) 2 y = r 2 − ( x − R ) 2 (1) The region lies between x = R + r and x = R − r . Sketch the solid region as shown in Figure 1. Calculate the volume using the method of cylindricalshell: V = ∫ a b 2 π x [ f ( x ) ] d x (2) Substitute R + r for a , R − r for b , and 2 r 2 − ( x − R ) 2 for [ f ( x ) ] in Equation (2). V = ∫ R − r R + r 2 π x 2 r 2 − ( x − R ) 2 d x = 4 π ∫ R − r R + r x r 2 − ( x − R ) 2 d x (3) Consider u = x − R (4) Differentiate both sides of the equation. d u = d x Rearrange Equation (4). u + R = x x = u + R Calculate the lower limit value of u using Equation (4). Substitute R + r for x in Equation (4). u = R + r − R = r Calculate the upper limit value of u using Equation (4). Substitute R − r for x in Equation (4). u = R − r − R = − r Apply lower and upper limits for u in Equation (3). Substitute u − R for x , u for ( r − R ) , and du for dx in Equation (3)

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 6.3, Problem 46E

To determine

To find: The volume of the solid (torus of Exercise 6.2.63) using cylindrical shell.

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Chapter 6.3, Problem 45E

Chapter 6.3, Problem 47E

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Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 7E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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