Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.3.141

Capitolo 7, Problema 7.3.141

In Exercises 139–142, find the length of each curve.
141. y = ln(cos(x)) from x = 0 to x = π/4.

Guida verificata passo dopo passo

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

Identify the function given: \[y = \ln(\cos(x))\] with the interval from \(x = 0\) to \(x = \frac{\pi}{4}\).

Find the derivative \(\frac{dy}{dx}\) using the chain rule: Since \(y = \ln(u)\) where \(u = \cos(x)\), then \[\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x)\]

Substitute \(\frac{dy}{dx} = -\tan(x)\) into the arc length formula: \[L = \int_0^{\frac{\pi}{4}} \sqrt{1 + (-\tan(x))^2} \, dx = \int_0^{\frac{\pi}{4}} \sqrt{1 + \tan^2(x)} \, dx\]

Use the trigonometric identity \(1 + \tan^2(x) = \sec^2(x)\) to simplify the integrand: \[L = \int_0^{\frac{\pi}{4}} \sqrt{\sec^2(x)} \, dx = \int_0^{\frac{\pi}{4}} |\sec(x)| \, dx\] Since \(\sec(x)\) is positive on \([0, \frac{\pi}{4}]\), this becomes \[L = \int_0^{\frac{\pi}{4}} \sec(x) \, dx\] The next step would be to evaluate this integral.

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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 length by summing infinitesimal line segments along the curve.

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Arc Length of Parametric Curves

Derivative of y = ln(cos(x))

To find the arc length, you need the derivative dy/dx. For y = ln(cos(x)), use the chain rule: dy/dx = -tan(x), since the derivative of ln(u) is 1/u * du/dx and d/dx[cos(x)] = -sin(x).

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Derivative of the Natural Exponential Function (e^x)

Integration Techniques for Arc Length

Evaluating the arc length integral often requires simplifying the integrand and applying appropriate integration methods, such as substitution or recognizing standard integral forms, to compute the exact length over the given interval.

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Percorso guidato

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Arc Length of Parametric Curves

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