Explanation Given: Curves as y = x 3 − x and x -axis. Formula Used: The area between curves is the area between the curve f ( x ) and curve g ( x ) on an interval [ a , b ] is given as follows A = ∫ a b | f ( x ) − g ( x ) | d x Calculation: Curve is given as follows y = x 3 − x And x -axis, y = 0 We need to first calculate the intersection points of the curve. Equate the two functions, we get x 3 − x = 0 x ( x 2 − 1 ) = 0 Thus, on solving for x , we get x = 0 or x 2 = 1 Thus, we have x = 0 , 1 , − 1 . Thus, the interval is [ − 1 , 1 ] . Area between the curves is calculate as follows A = ∫ 0 1 | ( x 3 − x ) − 0 | d x = ∫ 0 1 | ( x 3 − x ) | d x Let us calculate the integral without the absolutes. A = ∫ − 1 1 | ( x 3 − x ) | d x = ∫ − 1 0 ( x 3 − x ) d x + ∫ 0 1 ( x − x 3 ) d x Calculating the value of ∫ − 1 0 ( x 3 − x ) d x + ∫ 0 1 ( x − x 3 ) d x

6th Edition

ISBN: 9781119275565

Author: Hughes-Hallett

Publisher: WILEY

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Chapter 6.3, Problem 27P

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To find the exact area of the region bounded by the x -axis and the graph y=x3−x.

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Chapter 6.3, Problem 26P

Chapter 6.3, Problem 28P

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To find the exact area of the region bounded by the x -axis and the graph y = x 3 − x .

Solution Summary: The author calculates the area of the region bounded by the x -axis and the graph y=x3-x.

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To find the exact area of the region bounded by the x -axis and the graph y=x3−x.

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Chapter 6 Solutions