Explanation Given information: The given function is y = x 2 and the interval is [ 0 , 2 ] . Formula used: Assume that f ( x ) ≥ 0 and f ′ exists and continuous on the interval [ a , b ] . The surface area S of revolution about the x -axis over the interval is given by S = 2 π ∫ a b f ( x ) 1 + [ f ′ ( x ) ] 2 d x The reduction formula of the integral of sec n x is shown below. ∫ sec n x d x = sec n − 2 x tan x n − 1 + n − 2 n − 1 ∫ sec n − 2 d x + C , n ≠ 1 Calculation: Consider the given function. y = f ( x ) = x 2 Differentiate the above function with respect to x . f ′ ( x ) = 2 x The given interval is [ 0 , 2 ] . Here, a = 0 , b = 2 For 0 ≤ x ≤ 2 , x ≥ 0 and f ′ exists and continuous on [ 0 , 2 ] . Now calculate the surface area of revolution about x -axis over the interval [ 0 , 2 ] as shown below. S = 2 π ∫ a b f ( x ) 1 + [ f ′ ( x ) ] 2 d x = 2 π ∫ 0 2 x 2 1 + [ 2 x ] 2 d x = 2 π ∫ 0 2 x 2 1 + 4 x 2 d x Let us consider the substitution x = 1 2 tan u . So, d x = 1 2 sec 2 u d u and 1 + 4 x 2 = 1 + tan 2 u = sec u Limit will change as follows: x = 0 ⇒ 0 = 1 2 tan u ⇒ u = 0 x = 2 ⇒ 2 = 1 2 tan u ⇒ u = tan − 1 ( 4 ) So, the integral becomes 2 π 8 ∫ 0 tan − 1 ( 4 ) tan 2 u sec u sec 2 u d u = 2 π 8 ∫ 0 tan − 1 ( 4 ) ( sec 2 u − 1 ) sec

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ISBN: 9781319379421

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 8.2, Problem 41E

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The surface area of revolution about the x -axis over the interval [0,2] for the given function.

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Chapter 8.2, Problem 40E

Chapter 8.2, Problem 42E

Chapter 8 Solutions

CALCULUS 4E (HC) W/ ACHIEVE ACCESS

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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The surface area of revolution about the x -axis over the interval [ 0 , 2 ] for the given function.

Solution Summary: The author explains how to compute the surface area of revolution about the x -axis over the interval.

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To compute:

The surface area of revolution about the x -axis over the interval [0,2] for the given function.

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Chapter 8 Solutions

The surface area of revolution about the x -axis over the interval [ 0 , 2 ] for the given function.

Solution Summary: The author explains how to compute the surface area of revolution about the x -axis over the interval.

Videos

To compute: The surface area of revolution about the x -axis over the interval [0,2] for the given function.

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Chapter 8 Solutions

To compute:

The surface area of revolution about the x -axis over the interval [0,2] for the given function.