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Arc length of the natural logarithm Consider the curve y =

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 66E Chapter 7.4

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 66E

Arc length of the natural logarithm Consider the curve y = ln x.

a. Find the length of the curve from x =1 to x = a and call it L(a). (Hint: The change of variables  allows evaluation by partial fractions.)

b. Graph L(a).

c. As a increases, L(a)increases as what power of a?

Step-by-Step Solution:

Problem 66E

Arc length of the natural logarithm Consider the curve y = ln x.

a. Find the length of the curve from x = 1 to x = a and call it L(a).

b. Graph L(a).

c. As a increases, L(a) increases as what power of a?

Solution

Step 1

a. Find the length of the curve from x = 1 to x = a and call it L(a).

Here we have to find the length of the curve, y = ln x on [1,a]

Consider

Differentiating we get

The length of the curve is given by

Thus  …  (1)

Take

Differentiating we get,

When we get

When we get

Step 2 of 9

Chapter 7.4, Problem 66E is Solved
Step 3 of 9

Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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Arc length of the natural logarithm Consider the curve y =