Question

The length of \(x\) for \(y=\cosh (x)\) from \(x=0\) to \(x=2\)

Short answer

The arc length from \(x=0\) to \(x=2\) is \(\frac{e^2 - e^{-2}}{2}\).

Step-by-step solution

Understand the Definition of the Function

The hyperbolic cosine function is defined as \(y = \cosh(x) = \frac{e^x + e^{-x}}{2}\). We need to find the arc length of this function from \(x=0\) to \(x=2\).

Arc Length Formula for Hyperbolic Functions

The formula for the arc length \(L\) of a function \(y = f(x)\) from \(x=a\) to \(x=b\) is:\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \; dx \]For \(y = \cosh(x)\), we need \( \frac{dy}{dx} \).

Differentiate the Function

Find the derivative of \(y = \cosh(x)\). The derivative is \( \frac{dy}{dx} = \sinh(x) = \frac{e^x - e^{-x}}{2} \).

Substitute the Derivative into the Arc Length Formula

Substitute \( \frac{dy}{dx} = \sinh(x) \) into the formula:\[ L = \int_{0}^{2} \sqrt{1 + \sinh^2(x)} \; dx \]

Simplify the Integrand Using a Hyperbolic Identity

Use the identity \(1 + \sinh^2(x) = \cosh^2(x)\) to simplify the integrand:\[ L = \int_{0}^{2} \sqrt{\cosh^2(x)} \; dx = \int_{0}^{2} \cosh(x) \; dx \]

Integrate the Simplified Function

Since \( \int \cosh(x) \; dx = \sinh(x) \), we integrate:\[ L = \left[ \sinh(x) \right]_{0}^{2} \]

Evaluate the Definite Integral

Evaluate \( \sinh(x) \) at the bounds:\[ L = \sinh(2) - \sinh(0) \]\(\sinh(0) = 0\) and \(\sinh(2) = \frac{e^2 - e^{-2}}{2}\).So, the length is:\[ L = \frac{e^2 - e^{-2}}{2} \]

Calculate Numerically if Necessary

For practical purposes, you may wish to calculate a numerical approximation of the length:\[ L \approx 3.62686 \]

Key concepts

hyperbolic cosine

Hyperbolic functions are essential in various areas of mathematics, just like trigonometric functions in geometry. The hyperbolic cosine, denoted as \( \cosh(x) \), is one such important function. It is defined as:\[y = \cosh(x) = \frac{e^x + e^{-x}}{2}\]This formula might look familiar to the formula for the traditional cosine function but remember this is slightly different. The hyperbolic cosine represents the shape of a hanging cable, known as a catenary. It is even, symmetric about the y-axis, and always positive, similar to its trigonometric counterpart: the cosine function. Understanding this basic definition helps grasp more complex topics involving hyperbolic functions, including their applications in calculus and physics.

cosh(x)

When dealing with hyperbolic functions, \( \cosh(x) \) becomes a core element. It's not just a theoretical concept but also a practical one used in calculating arc lengths in hyperbolic geometry.Some quick properties of \( \cosh(x) \):- The graph of \( \cosh(x) \) is U-shaped, unlike the oscillating cosine.- \( \cosh(x) \) tends towards infinity as \( x \) gets larger, confirming that it grows exponentially.- The function is even, meaning \( \cosh(x) = \cosh(-x) \).Knowing these properties provides critical insight when integrating \( \cosh(x) \) or using it in calculus equations, such as calculating an arc length.

arc length integral formula

Calculating the arc length of a curve in calculus often utilizes integration techniques. For a function \( y = f(x) \) from \( x = a \) to \( x = b \), the arc length \( L \) is given by:\[L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx\]This formula finds the total length of a curve by integrating a segment's infinitesimal lengths over the entire interval.
For hyperbolic functions like \( y = \cosh(x) \), understanding this formula's derivation is crucial for solving related problems:

• First, determine \( \frac{dy}{dx} \), the derivative.
• Substitute the derivative into the integrand \( \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \).
• Use simplification techniques like hyperbolic identities to ease the integration process.

Applying these steps accurately is vital for finding precise arc lengths, which may be needed for engineering or physics calculations involving curves.

derivative sinh(x)

To calculate the arc length, as seen earlier, you need the derivative of \( \cosh(x) \). The derivative of \( \cosh(x) \) is the hyperbolic sine function, denoted by \( \sinh(x) \), which is defined as:\[\frac{dy}{dx} = \sinh(x) = \frac{e^x - e^{-x}}{2}.\]This function measures the rate of change of \( \cosh(x) \) and plays a significant role in integration processes involving hyperbolic functions:- \( \sinh(x) \) behaves similarly to the standard sine function in calculus contexts.- It is defined through the exponential function and grows rapidly with increasing \( x \).- Knowing \( \sinh(x) \)'s properties aids in simplifying integration problems, making it a cornerstone of many calculations involving hyperbolic arcs.By understanding how to derive and apply \( \sinh(x) \), you ensure accuracy in calculating more complex integrals and verify solutions in hyperbolic trigonometry and calculus.