Problem 6 - Calculus: Early Transcendentals

Question

Find the arc length of \(f(x)=\cosh x\) on \([0, \ln 2]\)

Step-by-Step Solution

Answer

The arc length is \(\frac{3}{4}\).

Step 1: Understand the Problem

We need to find the arc length of the function \(f(x) = \cosh x\) over the interval \([0, \ln 2]\). Recall that the arc length \(L\) of a function \(f(x)\) between two points \(a\) and \(b\) is given by the formula: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx \] where \( \frac{df}{dx} \) is the derivative of the function \(f(x)\).

Step 2: Find the Derivative of the Function

The function is \(f(x) = \cosh x\). The derivative of \(\cosh x\) is \(\sinh x\). Thus, \( \frac{df}{dx} = \sinh x \).

Step 3: Set Up the Integral for Arc Length

Substitute the derivative into the arc length formula: \[ L = \int_{0}^{\ln 2} \sqrt{1 + (\sinh x)^2} \, dx \]. Recall the identity \((\sinh x)^2 + 1 = (\cosh x)^2 \). Thus, the integral simplifies to: \[ L = \int_{0}^{\ln 2} \sqrt{(\cosh x)^2} \, dx = \int_{0}^{\ln 2} \cosh x \, dx \].

Step 4: Evaluate the Integral

Now, solve the integral \(\int_{0}^{\ln 2} \cosh x \, dx\). The antiderivative of \(\cosh x\) is \(\sinh x\). Thus, \[ \int \cosh x \, dx = \sinh x \]. Evaluate this from 0 to \(\ln 2\): \[ L = \sinh(\ln 2) - \sinh(0) \].

Step 5: Simplify the Result

Calculate \(\sinh(\ln 2)\) and \(\sinh(0)\): \(\sinh(\ln 2) = \frac{e^{\ln 2} - e^{-\ln 2}}{2} = \frac{2 - \frac{1}{2}}{2} = \frac{3}{4} \). Since \(\sinh(0) = 0\), the arc length \(L = \frac{3}{4}\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogous to the trigonometric functions but are based on hyperbolas instead of circles. They are widely used in calculus and other areas of mathematics due to their particular properties and applications. The most common hyperbolic functions are the hyperbolic sine, denoted as \( \sinh x \), and hyperbolic cosine, denoted as \( \cosh x \).
Many of the properties of hyperbolic functions are similar to those of circular functions. For instance:

• The derivative of \( \sinh x \) is \( \cosh x \)
• The derivative of \( \cosh x \) is \( \sinh x \)

The relationship between \( \sinh x \) and \( \cosh x \) is crucial in solving problems involving these functions, particularly in calculus when finding arc lengths, surface areas, and other quantities.

Calculus Integration

Integration is a fundamental operation in calculus, just like differentiation. It is the process of finding the integral of a function, which represents the accumulation of quantities, such as areas under curves. In the context of arc length, integration helps us calculate the length of a curve over a given interval.
When finding the arc length of a function using integration, you first need to calculate the function's derivative. This derivative is then used in the arc length formula. The actual calculation involves evaluating the definite integral of a function over a specified range. This integral sums up the infinitesimally small segments of the curve, producing the total length.
Being comfortable with integration techniques is crucial for solving arc length problems, as these integrals can sometimes be complex and require a solid understanding of calculus principles.

Arc Length Formula

The arc length formula is essential when determining the length of a curve described by a function. If you have a continuous and differentiable function, like \( f(x) \), spanning an interval \([a, b]\), the arc length \( L \) is given by:\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx \]This formula derives from partitioning the curve into straight-line segments and then calculating their lengths. As these segments become infinitesimally small, their sum approaches the actual arc length.
For the problem at hand, \( f(x) = \cosh x \), which required finding \( \frac{df}{dx} \), showing the importance of applying the formula accurately to find the length between two points on the curve.

Sinh and Cosh Functions

The sinh and cosh functions are hyperbolic sine and cosine functions, respectively, vital in solving various mathematical problems, including arc length calculations. They are defined as:

• \( \sinh x = \frac{e^x - e^{-x}}{2} \)
• \( \cosh x = \frac{e^x + e^{-x}}{2} \)

The relationship \((\sinh x)^2 + 1 = (\cosh x)^2\) is often used to simplify expressions involving these functions, such as in the integral for the arc length of \( \cosh x \).
In the arc length calculation, knowing the identities and derivatives of sinh and cosh helps simplify the integration process and, in turn, the calculations needed to find the arc length. Understanding these functions' definitions and properties is key in many applied mathematics problems, showing their integral role in calculus and beyond.