Question

Find the arc length of the function on the given interval. \(f(x)=\ln (\cos x)\) on \([0, \pi / 4]\)

Short answer

The arc length is \( \ln(\sqrt{2} + 1) \).

Step-by-step solution

Know the Formula for Arc Length

The formula for the arc length of a function \( f(x) \) on an interval \([a, b]\) is given by \[L = \int_{a}^{b} \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx\]We will apply this formula to find the arc length of \( f(x)=\ln(\cos x) \) on \([0, \pi/4]\).

Find the Derivative of the Function

Determine the derivative \( \frac{df}{dx} \) of the function. For \( f(x) = \ln(\cos x) \), the derivative is \[\frac{df}{dx} = \frac{-\sin x}{\cos x} = -\tan x\]

Set up the Integrand

Substitute the derivative into the integrand of the arc length formula. This gives: \[L = \int_{0}^{\pi/4} \sqrt{1 + (-\tan x)^2} \, dx\]Since \( \tan^2 x + 1 = \sec^2 x \), the integrand simplifies to: \[L = \int_{0}^{\pi/4} \sec x \, dx\]

Evaluate the Integral

The integral of \( \sec x \) is known to be \( \ln|\sec x + \tan x| + C \). Evaluating this definite integral from 0 to \( \pi/4 \), we have:\[L = \left[ \ln|\sec x + \tan x| \right]_{0}^{\pi/4}\]Substitute \( x = \pi/4 \):\[ \sec(\pi/4) + \tan(\pi/4) = \sqrt{2} + 1 \]Substitute \( x = 0 \): \[ \sec(0) + \tan(0) = 1 + 0 \]Thus, \[L = \ln(\sqrt{2} + 1) - \ln(1) = \ln(\sqrt{2} + 1)\]

Simplify the Final Answer

Since the natural logarithm of 1 is 0, the arc length simplifies to \[L = \ln(\sqrt{2} + 1) \] Therefore, the arc length of the function \( f(x) = \ln(\cos x) \) from 0 to \( \pi/4 \) is \( \ln(\sqrt{2} + 1) \).

Key concepts

Integral Calculus

Integral calculus is a fundamental aspect of mathematical analysis that deals with the concept of integration. Integration is essentially the inverse operation of differentiation and is used to find quantities such as areas, volumes, and in our case, arc lengths. It provides a way to sum an infinite number of infinitesimally small quantities, resulting in a finite number.
Integration is crucial in mathematics because it allows us to solve problems involving accumulation of quantities. For example, in physics, it helps to determine the total distance traveled by an object from its velocity function. In our arc length problem, integration allows us to calculate the length of a curve by summing up the infinitesimal arc segments over the interval [0, \pi/4].
When approaching integration, it is essential to:

• Identify the function to integrate.
• Determine the interval over which to perform the integration.
• Apply techniques such as substitution or integration by parts when necessary.

In our problem, we use definite integration to sum up the arc length from 0 to \(\pi/4\).

Derivative

The derivative of a function measures how that function changes as its input changes. It provides the rate at which one quantity changes with respect to another. Derivatives are a basic tool in calculus and are often symbolized by \(\frac{df}{dx}\), which stands for the derivative of function \(f\) with respect to \(x\).
In the context of finding arc length, calculating the derivative is crucial, as it helps determine how the function's slope changes along the interval. This information is necessary to set up the integrand in the arc length formula. For the function \(f(x) = \ln(\cos x)\), the derivative is \(\frac{df}{dx} = -\tan x\).
The derivative involves various rules and techniques:

• The power rule for basic polynomials.
• Product and quotient rules for functions multiplied or divided by each other.
• The chain rule for composite functions.

In our exercise, knowing how to find the derivative of \(\ln(\cos x)\) was key to proceeding with solving the problem.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in calculus, because they relate the angles of a triangle to the lengths of its sides. Some primary trigonometric functions include sine, cosine, and tangent. These functions are periodic and are especially important when dealing with wave phenomena or oscillations.
In our exercise, the function \(f(x) = \ln(\cos x)\) involves the cosine function, which oscillates between -1 and 1. The behavior of trigonometric functions, such as \(\cos x\) and \(-\tan x\), is essential in our calculation of the arc length. Using the trigonometric identity \(\tan^2 x + 1 = \sec^2 x\) helps simplify the integrand in the arc length formula.
Key trigonometric identities to remember:

• \(\sin^2 x + \cos^2 x = 1\)
• \(1 + \tan^2 x = \sec^2 x\)
• \(\sec x = \frac{1}{\cos x}\)

These identities often simplify complex expressions and make calculations more manageable in integral calculus.

Definite Integral

The concept of a definite integral extends the idea of an indefinite integral by providing limits to the integration process. It represents the net area under a curve within a specified interval, such as the interval [0, \pi/4] in our problem.
In practical terms, a definite integral calculates the accumulated value of a function as it varies over the specified limits. In our exercise, the definite integral is crucial for finding the arc length as it adds up all infinitesimal lengths along the curve from \(x = 0\) to \(x = \pi/4\).
Steps for evaluating a definite integral include:

• Identify the function to integrate.
• Determine the limits of integration.
• Calculate the integral of the function, apply the integration limits, and evaluate.

For \(\int_{0}^{\pi/4} \sec x \, dx\), we found out that evaluating between the limits gives the arc length \(L = \ln(\sqrt{2} + 1)\), which is the net length of the logarithmic cosine curve on the interval provided.