Explanation Given: The equation is y = ( 4 − x ) ln x ; 1 ≤ x ≤ 7 . Theorem used: If f and g are continuous and f ( x ) ≥ g ( x ) over the interval [ a , b ] , then the area bounded by y = f ( x ) and y = g ( x ) for a ≤ x ≤ b is given exactly by A = ∫ a b [ f ( x ) − g ( x ) ] d x . Calculation: Use online graphing calculator to sketch the graph of the given functions y = ( 4 − x ) ln x in the interval [ 1 , 7 ] is as shown below in Figure 1. From Figure 1, it is observed that the graph of f ( x ) intersects the x -axis at the point (4,0). The graph of f ( x ) ≥ 0 on [ 1 , 4 ] and f ( x ) ≤ 0 on [ 4 , 7 ] . By the above mentioned theorem, the area of the shaded region is computed as follows. A = ∫ 1 4 ( ( 4 − x ) ln x − 0 ) d x + ∫ 4 7 ( 0 − ( 4 − x ) ln x ) d x = ∫ 1 4 ( 4 − x ) ln x d x − ∫ 4 7 ( 4 − x ) ln x d x Use online graphing calculator to calculate the integral ∫ ( 4 − x ) ln x d x as follows, ∫ ( 4 − x ) ln x d x = − 1 2 x 2 ln x + x 2 − 3 x 2 4 + 4 x ln x − 4 x Apply the upper and lower limit of the integral ∫ 1 4 ( 4 − x ) ln x d x

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 6.3, Problem 69E

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To sketch: The graph of the equation y=(4−x)lnx and find the area between the curve and the x axis over the interval [1,7].

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Chapter 6.3, Problem 68E

Chapter 6.3, Problem 70E

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Chapter 6 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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The graph of the equation y = ( 4 − x ) ln x and find the area between the curve and the x axis over the interval [ 1 , 7 ] .

Solution Summary: The graph of the equation y=(4-x)mathrmlnx and find the area between the curve and the x axis is 10.87

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To sketch: The graph of the equation y=(4−x)lnx and find the area between the curve and the x axis over the interval [1,7].

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Chapter 6 Solutions