Problem 452 - Calculus Volume 1

Question

For the following exercises, find the requested arc lengths. The length of \(x\) for \(y=\cosh (x)\) from \(x=0\) to \(x=2\)

Step-by-Step Solution

Answer

The arc length is \( \sinh(2) \).

Step 1: Understand the Problem

We need to find the arc length of the function \( y = \cosh(x) \) from \( x = 0 \) to \( x = 2 \). The arc length \( L \) of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is calculated using the formula \( L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \).

Step 2: Compute the Derivative

Calculate the derivative of \( y = \cosh(x) \), which is \( \frac{dy}{dx} = \sinh(x) \). This derivative will be used in the arc length formula.

Step 3: Set Up the Arc Length Integral

Substitute the derivative into the arc length formula. We have:\[ L = \int_{0}^{2} \sqrt{1 + \sinh^2(x)} \, dx \].

Step 4: Simplify Using Trigonometric Identity

Use the hyperbolic identity \( \cosh^2(x) = 1 + \sinh^2(x) \) to simplify the integral:\[ L = \int_{0}^{2} \cosh(x) \, dx \].

Step 5: Integrate the Simplified Expression

Evaluate the integral of \( \cosh(x) \). The integral of \( \cosh(x) \) is \( \sinh(x) \), so:\[ L = \left[ \sinh(x) \right]_{0}^{2} \].

Step 6: Calculate the Definite Integral

Substitute the limits into the evaluated integral:\[ L = \sinh(2) - \sinh(0) \].We know \( \sinh(0) = 0 \), so:\[ L = \sinh(2) \].

Step 7: Evaluate \( \sinh(2) \)

Calculate \( \sinh(2) = \frac{e^2 - e^{-2}}{2} \) using the definition of the hyperbolic sine function. This gives us the arc length.

Key Concepts

Hyperbolic Functions

Hyperbolic functions such as the hyperbolic cosine, denoted as \( \cosh(x) \), and the hyperbolic sine, denoted as \( \sinh(x) \), are analogues of the trigonometric functions cosine and sine but for a hyperbola. They appear frequently in many areas of mathematics, including calculus, geometry, and the theory of complex numbers.
One key feature of these functions is their definitions in terms of exponential functions:

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

These functions are helpful in describing the shape of hanging cables and in solving certain differential equations. They also have distinct derivatives, with the derivative of \( \cosh(x) \) being \( \sinh(x) \) and the derivative of \( \sinh(x) \) being \( \cosh(x) \), reminiscent of the relationships seen between sine and cosine.
Understanding these functions is fundamental to proceeds with solving problems involving hyperbolic identities and calculations.

Definite Integral

A definite integral computes the accumulation of quantities, which in many instances can be visualized as the area under a curve between two points. In the context of arc length calculation, it helps aggregate the infinitesimal lengths of curve segments over an interval.
In this problem, we need to compute the definite integral of the expression derived from the arc length formula:
\[ L = \int_{0}^{2} \cosh(x) \, dx \]This integral calculation incorporates the simplification process achieved using hyperbolic identities, leading to an easier evaluation.
Solving the integral eventually involves calculating a closed form, meaning you derive an explicit number instead of another function. It's useful because it gives the exact length of the arc over the interval from \(x=0\) to \(x=2\). This result empowers us to understand measurements in terms of hyperbolic curves effectively.

Derivative Calculation

In calculus, finding the derivative of a function is essential for understanding the function's behavior, including its rate of change and slope at any given point. For hyperbolic functions, calculating derivatives is a straightforward but crucial task.
For the given function \( y = \cosh(x) \), the derivative is found as follows:\[ \frac{dy}{dx} = \sinh(x) \]This derivative is vital for our arc length formula since it helps in determining the local slope of the hyperbolic curve.
The derivative calculation directly influences the arc length since it’s incorporated within the square root in the integral. Without correctly computing it, subsequent steps like setting up or transforming the integral wouldn’t be possible. Using derivative rules simplifies expressions effectively, aiding in preparing functions for integration or additional manipulation.

Hyperbolic Identity

Hyperbolic identities, like trigonometric identities, are fundamental relationships between hyperbolic functions that simplify mathematical expressions. For this exercise, we utilize one such relationship:
\[ \cosh^2(x) = 1 + \sinh^2(x) \]This identity mirrors the well-known Pythagorean identity in trigonometry. It plays a crucial role by transforming the integrand \( \sqrt{1 + \sinh^2(x)} \) into \( \cosh(x) \), dramatically simplifying the integration process needed to determine the arc length.
Using hyperbolic identities reduces the complexity of calculations. Instead of dealing with an unwieldy square root, this identity allows swift and effective integration. Recognizing and applying such identities is a critical skill, especially when encountering arc length calculations in hyperbolic curves.