Explanation 1) Concept: Formula: The area A of the region bounded by the curves y = f ( x ) , y = g ( x ) and the lines x = a a n d x = b is A = ∫ a b f x - g x d x f x - g x = f x - g x w h e n f x ≥ g ( x ) g x - f x w h e n g x ≥ f ( x ) 2) Given: y = cos ⁡ π x and y = 4 x 2 - 1 3) Calculation: The point of intersection occurs when both equations are equal to each other, that is, cos ⁡ π x = 4 x 2 - 1 As - 1 ≤ cos ⁡ θ ≤ 1 So at point of intersection, cos ⁡ π x = 4 x 2 - 1 That is 4 x 2 - cos ⁡ π x - 1 = 0 By quadratic formula thus x = ± 0 - 4 ( 4 ) ( - cos ⁡ π x - 1 ) 2 ( 4 ) = ± 16 ( cos ⁡ π x + 1 ) 8 = ± ( cos ⁡ π x + 1 ) 2 Thus by trial and error (in view of the graph) points of intersection are at x = - 1 2 and x = 1 2 . The region is sketched in the following figure. We may also directly use the graph generated by some graphing utility to find the points of intersections. Here, cos ⁡ π x ≥ 4 x 2 - 1 when - 1 2 ≤ x ≤ 1 2

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5.1, Problem 19E

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Sketch the region and find the enclosed area.

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Chapter 5.1, Problem 18E

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Chapter 5 Solutions

Calculus (MindTap Course List)

Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Prob. 3E Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 10E

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To: Sketch the region and find the enclosed area.

Solution Summary: The author explains that the point of intersection occurs when both equations are equal to each other.