Question

Arc length Find the length of the curve $y=\ln (\cos x)$ $0 \leq x \leq \pi / 3$

   Arc length Find the length of the curve $y=\ln (\cos x)$
$0 \leq x \leq \pi / 3$

Thomas Calculus

Thomas Calculus

George B. Thomas,… 13th Edition

Chapter 8, Problem 43

Instant Answer

Solved by Expert Nick Johnson

07/01/2023

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The arc length \( L \) is given by: \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] Show more…

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Arc length Find the length of the curve $y=\ln (\cos x)$ $0 \leq x \leq \pi / 3$

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Key Concepts

Integration of Trigonometric Functions

After simplifying the integrand using trigonometric identities, the integration step may involve standard integrals of trigonometric functions. For instance, integrating functions like sec x involves specialized techniques and formulas, which are essential in evaluating the arc length integral accurately.

Trigonometric Identities

Key identities, such as 1 + tan²x = sec²x, simplify the process of arc length integration. These identities allow the transformation of the integrand into a more manageable form, often converting it into an elementary function that can be directly integrated.

Differentiation of Logarithmic and Trigonometric Functions

In problems involving a function like ln(cos x), one must differentiate using both logarithmic differentiation and the derivatives of trigonometric functions. This typically involves recognizing that the derivative of ln(u) is u'/u, and using the known derivative of cosine, which is -sin x, to obtain the complete derivative.

Derivative of Composite Functions Using the Chain Rule

When differentiating functions composed of other functions, such as logarithmic functions of trigonometric functions, the chain rule is used. This rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function, ensuring accurate calculation of rates of change required in arc length computations.

Arc Length Formula

This concept refers to calculating the length of a curve defined by a function y = f(x) over an interval [a, b]. The formula, L = ?[a to b] ?(1 + (dy/dx)²) dx, is foundational in calculus for measuring distances along curves by integrating small segments that approximate the true length.

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