Explanation Given information: Horizontal line y = c is intersecting the curve y = 8 x − 27 x 3 . Show the intersection of curve and the horizontal line as shown in Figure 1. Calculation: Show the equation of line as follows: y = c (1) Show the equation of the curve as follows: y = 8 x − 27 x 3 (2) Equating Equation (1) and Equation (2). c = 8 x − 27 x 3 (3) Refer to Figure 1. The region R 1 lies between x = 0 and x = a and the region R 2 lies between x = a and x = b . Consider the areas of the shaded regions are equal. R 1 = R 2 (4) Here, R 1 is the area of the shaded region within the limits x = 0 and x = a and R 2 is the area of the shaded region within the limits x = a and x = b . Modify Equation (4). ∫ 0 a ( c − ( 8 x − 27 x 3 ) ) d x = ∫ a b ( ( 8 x − 27 x 3 ) − c ) d x ( c x − 8 x 2 2 + 27 x 4 4 ) 0 a = ( 8 x 2 2 − 27 x 4 4 − c x ) a b (5) Substitute 8 x − 27 x 3 for c in Equation (5)

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6, Problem 3P

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To find: The number c such that the areas of the shaded régions are equal.

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Chapter 6, Problem 2P

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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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