Explanation Given information: Theequation of the curve is x 2 3 + y 2 3 = 1 , 0 ≤ y ≤ 1 The curve is symmetric about the y -axis from x = − 1 to x = 1 , Hence, use the portion of the curve from x = 0 to x = 1 Therefore, take the area of curve which is bounded between x = 0 and x = 1 Calculation: Show the equation of the curve. x 2 3 + y 2 3 = 1 (1) Rewrite the Equation (1) as follows x 2 3 + y 2 3 = 1 y 2 3 = 1 − x 2 3 y = ( 1 − x 2 3 ) 3 2 (2) Calculate the area of the surface obtained by rotating the curve about y -axis using the relation: S = ∫ a b 2 π x 1 + ( d y d x ) 2 d x (3) Here, S is the area of the surface obtained by rotating the curve about y -axis. Differentiate both sides of Equation (2) with respect to x . d y d x = d d x [ ( 1 − x 2 3 ) 3 2 ] = 3 2 ( 1 − x 2 3 ) 3 2 − 1 ( − 2 3 x 2 3 − 1 ) = ( 1 − x 2 3 ) 1 2 ( − x − 1 3 ) = − 1 − x 2 3 x − 1 3 Substitute − 1 − x 2 3 x − 1 3 for d y d x , 0 for a, and 1 for b in Equation (3)

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ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8.2, Problem 16E

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To find: The area of the surface obtained by rotating the curve about y-axis.

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Chapter 8.2, Problem 15E

Chapter 8.2, Problem 17E

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The area of the surface obtained by rotating the curve about y -axis.

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Chapter 8 Solutions