Explanation Given information: Given to find the exact areabetween y = ln x and y = ln ( x 2 ) for 1 ≤ x ≤ 2 . Calculation: The exact area under the graph is given by the definite integral: A = ∫ y 2 − y 1 = ∫ 1 2 [ ln ( x 2 ) − ln x ] d x A = ∫ 1 2 ( 2 ln x − ln x ) d x A = ∫ 1 2 ln x d x Solving the indefinite integral using integration by parts: According to integration by parts, ∫ u v ' = u v − ∫ u ' v Here, u = ln x , v ' = 1 Consider, u = ln x Differentiating both sides with respect to x , u ' = d d x ( ln x ) u ' = 1 x Consider, v ' = 1 Integrating both sides, ∫ v ' = ∫ 1 d x v = x

6th Edition

ISBN: 9781119267874

Author: Hughes-Hallett

Publisher: WILEY

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

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Chapter 6.7, Problem 27P

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Chapter 6.7, Problem 26P

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To find the exact area.

Solution Summary: The author explains how the exact area under the graph is given by the definite integral using integration by parts.

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Chapter 6 Solutions