Question

Show that the arc length of the catenary \(y=\cosh x\) over the interval \([0, a]\) is \(L=\sinh a\).

Short answer

Answer: The arc length of the catenary \(y = \cosh x\) over the interval \([0, a]\) is \(L = \sinh a\).

Step-by-step solution

Find the derivative of y with respect to x

Given the function \(y = \cosh x\), we can find its derivative \(\frac{dy}{dx}\) using the following rule: \(\frac{d}{dx} (\cosh x) = \sinh x\) So, we have: \(\frac{dy}{dx} = \sinh x\)

Substitute the derivative into the formula for arc length

Now we can substitute the derivative of \(y\) with respect to \(x\) into the arc length formula: \(L = \int_0^a \sqrt{1 + (\frac{dy}{dx})^2} dx = \int_0^a \sqrt{1 + (\sinh x)^2} dx\)

Simplify the expression under the square root

Note that using the identity \(\cosh^2 x - \sinh^2 x = 1\), we can rewrite the expression under the square root as follows: \(\sqrt{1 + (\sinh x)^2} = \sqrt{\cosh^2 x} = \cosh x\) Thus, the arc length formula becomes: \(L = \int_0^a \cosh x dx\)

Integrate and find the arc length

Now, we can integrate \(\cosh x\) over the interval \([0, a]\): \(L = \int_0^a \cosh x dx = [\sinh x]_0^a = \sinh a - \sinh 0 = \sinh a - 0 = \sinh a\) Therefore, the arc length of the catenary \(y = \cosh x\) over the interval \([0, a]\) is \(L = \sinh a\).

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogues of the trigonometric functions but for the hyperbola, much like trigonometric functions are for the circle. Two primary hyperbolic functions are hyperbolic cosine (\( \cosh x \) ) and hyperbolic sine (\( \sinh x \) ). These functions are defined using exponential functions:

• The hyperbolic cosine, \( \cosh x = \frac{e^x + e^{-x}}{2} \), represents the curve of a hanging cable or chain, known as a catenary.
• The hyperbolic sine, \( \sinh x = \frac{e^x - e^{-x}}{2} \), captures the shape variance compared to straight lines.

Given their relationship in formulas and identities, \( \cosh x \) and \( \sinh x \) possess identities similar to \( \sin x \) and \( \cos x \). One important identity used in calculations is \( \cosh^2 x - \sinh^2 x = 1 \). This mirrors the Pythagorean identity seen in trigonometry and is essential when simplifying expressions in calculus problems related to hyperbolic functions.

Integration

Integration is a crucial concept in calculus used to find quantities that add up over a range, such as area under curves, accumulated values, and in this case, arc length. To integrate a function mathematically means to determine the indefinite integral, or antiderivative, of a function.
In our context, we integrate the function \( \cosh x \) over a given interval using the fundamental theorem of calculus. The formula for integration that we use here is:

• For a given function \( f(x) \), the indefinite integral is \( \int f(x) \, dx \).
• To find the definite integral from \( a \) to \( b \), you evaluate the antiderivative at the two boundaries and subtract: \( \left[F(b) - F(a)\right] \).

When calculating the arc length of \( y = \cosh x \) over \([0, a]\), we use the fact that the antiderivative of \( \cosh x \) is \( \sinh x \).
The definite integral is evaluated by calculating \( \sinh x \) at \( a \) and then subtracting the value at \( 0 \). This solution seamlessly connects integration with arc length computation.

Derivative

Understanding derivatives involves knowing how functions change as their input changes. Derivatives help us understand the rate of change of a function.
In this exercise, when we first determine the derivative of \( y = \cosh x \), we get \( \frac{dy}{dx} = \sinh x \). This reflects how \( y \) changes with respect to \( x \). To find this derivative, we use a basic rule of differentiation for hyperbolic functions, which follows a similar process as differentiating trigonometric functions:

• Using the rule, \( \frac{d}{dx} (\cosh x) = \sinh x \).

The calculated derivative feeds directly into the arc length formula.
By using derivatives, we set the groundwork for further steps in calculus, such as determining geometry-related measures like arc lengths, which tell us about the curve's actual path when assessing the hyperbolic shape.

Arc Length Formula

The arc length formula gives us a straightforward way to calculate the distance along a curve over a specific interval.
For the function \( y = \cosh x \), we use the arc length formula given by:

• \( L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \ dx \)

In our particular problem, the derivative is \( \sinh x \), which substitutes into the formula, simplifying the integrand using the identity \( \cosh^2 x - \sinh^2 x = 1 \).
This step is crucial as it turns what could be a complex integral into a simpler one by showing that:

• \( \sqrt{1 + (\sinh x)^2} = \cosh x \)

The arc length \( L \) over the interval \([0, a]\) simplifies to the integral of \( \cosh x \), yielding \( L = \sinh a \). Understanding this formula allows students to relate the hyperbolic function and its derivative directly to physical properties such as the length of a catenary.