Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.R.108

Chapter 8, Problem 8.R.108

108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].

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Recall the formula for the arc length of a curve defined by a function \( y = f(x) \) on the interval \( [a, b] \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]

Identify the function and interval: here, \( y = \ln(x) \) and the interval is \( [1, e^2] \).

Compute the derivative \( \frac{dy}{dx} \) of \( y = \ln(x) \): \[ \frac{dy}{dx} = \frac{1}{x} \]

Substitute \( \frac{dy}{dx} \) into the arc length formula to get: \[ L = \int_1^{e^2} \sqrt{1 + \left(\frac{1}{x}\right)^2} \, dx = \int_1^{e^2} \sqrt{1 + \frac{1}{x^2}} \, dx \]

Simplify the integrand inside the square root: \[ \sqrt{1 + \frac{1}{x^2}} = \sqrt{\frac{x^2 + 1}{x^2}} = \frac{\sqrt{x^2 + 1}}{x} \] So the integral becomes: \[ L = \int_1^{e^2} \frac{\sqrt{x^2 + 1}}{x} \, dx \] Next, set up this integral for evaluation using an appropriate substitution or method.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the distance along the curve by summing infinitesimal line segments, accounting for both horizontal and vertical changes.

Derivative of the Natural Logarithm Function

For y = ln(x), the derivative dy/dx is 1/x. Understanding this derivative is essential to substitute into the arc length formula, as it determines the slope of the curve at each point, influencing the length calculation.

Definite Integration over a Given Interval

Evaluating the arc length requires computing a definite integral from x = 1 to x = e^2. This involves integrating the function √(1 + (1/x)^2) over the interval, which may require techniques such as substitution or recognizing standard integral forms.

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