Explanation Given information: Two curves x = sin y & x = 1 − cos y and we have to find the area over the interval − π 2 ≤ y ≤ π 2 Concept and Formula Used: We have to determine which graph is on the right and then we find the point of intersection of the two curves and find the area using the formula ∫ a b [ f ( y ) − g ( y ) ] d y where f ( y ) is the right curve and g ( y ) is the left curve. ∫ cos x d x = sin x + C , ∫ sin x d x = − cos x + C Calculation: The graph of the given equation is represented by From the graph we found that the bounded region is horizontal simple. From the graph the region is divided into two parts. First part is from − π 2 to 0 in which right curve is x = 1 − cos y and the curve at the left is x = sin y . In the second region i.e. from 0 to π 2 in which right curve is x = sin y and the curve at the left is x = 1 − cos y . The area of the region is given by A=A 1 + A 2 A 1 = ∫ a b [ f ( y ) − g ( y ) ] d y where f ( y ) = 1 − cos y & g ( y ) = sin y and a = − π 2 , b = 0 . So, find the area A 1 . A 1 = ∫ − π 2 0 [ ( 1 − cos y ) − sin y ] d y = ∫ − π 2 0 [ 1 − cos y − sin y ] d y = [ y ] − π 2 0 − [ sin y ] −

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.1, Problem 22E

To determine

The area of the shaded region between the curves x=siny and x=1−cosy in the interval 0≤y≤π2.

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Chapter 6.1, Problem 21E

Chapter 6.1, Problem 23E

Chapter 6 Solutions

Calculus - Standalone book

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The area of the shaded region between the curves x = sin y and x = 1 − cos y in the interval 0 ≤ y ≤ π 2 .

Solution Summary: The graph of the given equation is represented by two curves x=mathrmsiny and 2sq.units. The bounded region is horizontal simple.

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The area of the shaded region between the curves x=siny and x=1−cosy in the interval 0≤y≤π2.

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Chapter 6 Solutions