Question

Show that to find the length of an arc of the parabola \(y=\) \(x^{2}\) one needs to determine the area under the hyperbola \(y^{2}-4 x^{2}=1\)

Short answer

\(\frac{dy}{dx} = \frac{-8x}{2y}\) Now let's substitute \(x = \frac{1}{2}\sinh{u}\) and \(y = \cosh{u}\): \(\frac{dy}{du} = \frac{-8(\frac{1}{2}\sinh{u})}{2\cosh{u}}\) Simplifying and integrating: \(s = \int -\sinh{u} du\) #tag_title#Step 3: Connect the arc length of the parabola to the area under the hyperbola#tag_content#Now we have the expression for the arc length of the parabola in terms of the hyperbolic function. The area under the curve of the hyperbola \(y^{2} - 4x^{2} = 1\) can be found using integration: \(Area = \int_a^b y dx\) Now, we know that \(y = \cosh{u}\) and \(dx = \cosh{u} du\), so substitute and integrate: \(Area = \int_c^d \cosh^2{u} du\) Now making use of the property \(\cosh^2{u} - \sinh^2{u} = 1\) and the fact that \(-\sinh{u}\) is the derivative of the arc length: \(Area = -\int_c^d \sinh{u} du\) Comparing this expression with the arc length expression, we see that they are equal in terms of integration. Therefore, the area under the hyperbola is equal to the arc length of the parabola. #Answer_sentence# The area under the hyperbola \(y^2 - 4x^2 = 1\) is equal to the arc length of the parabola \(y = x^2\).

Step-by-step solution

Derive the expression for arc length of the parabola

First, let's find the expression for the arc length of the parabola \(y=x^{2}\). The formula for the arc length in Cartesian coordinates is given by: \(s = \int_a^b \sqrt{1+(\frac{dy}{dx})^2} dx\) where \(a\) and \(b\) are the x-coordinates of the endpoints of the arc. Now let's find \({dy}/{dx}\) for the given parabola \(y=x^2\): \(\frac{dy}{dx} = 2x\) Now, substitute this into the arc length formula: \(s = \int_a^b \sqrt{1+(2x)^2} dx\)

Transform the expression for arc length of the parabola using hyperbolic function

Let's introduce the hyperbolic function to transform the expression. We know that \(\cosh^2{u} - \sinh^2{u} = 1\), so we can substitute \(x= \frac{1}{2} \sinh{u}\): \(s = \int \sqrt{1+(2(\frac{1}{2}\sinh{u}))^2} (\cosh{u}) du\) Simplify the integrand: \(s = \int \sqrt{1+(\sinh^2{u})} \cosh{u} du\) Then, substituting the original definition of hyperbolic functions using exponentials, we have: \(s = \int \sqrt{\cosh^2{u} - \sinh^2{u}}\cosh{u} du\) Finally, using the expression for the hyperbola \(y^{2} - 4x^{2} = 1\), we can find the differential for \(y\):

Key concepts

Parabola

A parabola is a U-shaped curve that is defined mathematically as the set of all points equidistant from a point called the focus and a line called the directrix. The simplest form of a parabola is described by the equation \(y = x^2\). This is a symmetric curve, with its vertex at the origin and opening upwards.

For the equation \(y = x^2\), the gradient or the slope at any given point is determined by taking the derivative, which results in \(\frac{dy}{dx} = 2x\). This derivative represents how steep the parabola is at any point \(x\).

When finding the arc length, we're looking for the length of a segment of this curve between two points. This requires calculus, as the curve cannot simply be measured using a ruler. To do this, we use the formula for arc length, which involves integrating the square root of \(1 + (\frac{dy}{dx})^2\) over the interval specified.

Hyperbola

A hyperbola is a type of conic section, similar to an ellipse, but instead of being closed, it consists of two separate curves, or "branches." It is defined as the set of all points where the absolute difference of the distances to two fixed points called the foci is constant.

The equation \(y^2 - 4x^2 = 1\) describes a hyperbola in standard form. This equation separates into two curves represented by this expression, revealing the hyperbola's distinctive shape.

In the context of finding arc lengths, the hyperbola comes in handy when considering transformations, particularly when using hyperbolic functions. The transformation of the arc length integral of a parabola into the form involving a hyperbola can simplify calculations. Substituting variables related to the hyperbola's properties can help solve the arc length more effectively.

Hyperbolic Functions

Hyperbolic functions, including \(\sinh(u)\) and \(\cosh(u)\), are mathematical functions that offer parallels to the regular trigonometric functions \(\sin\) and \(\cos\). They arise naturally in many contexts, including the study of hyperbolas and hyperbolic geometry.

These functions are defined using exponential functions as follows:

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

Hyperbolic functions satisfy the identity \(\cosh^2(u) - \sinh^2(u) = 1\), which is reminiscent of the Pythagorean identity for sine and cosine. This property makes them useful for substituting variables in integral calculus, facilitating the transformation of problems into more workable forms.

In the scenario of finding the arc length of a parabola, substituting \(x = \frac{1}{2} \sinh(u)\) re-expresses the original integral related to the parabola in a form that involves hyperbolic trigonometric identities, simplifying the integral and making it more approachable.

Integral Calculus

Integral calculus is a branch of mathematics concerned with the concept of integrals. While differential calculus cuts and slices to find the rate of change, integral calculus stitches back together those slices to find accumulative totals, like area and lengths.

When calculating arc length, integral calculus provides a way to sum up infinitely small pieces of the curve to find the total length. The formula \(s = \int_a^b \sqrt{1+(\frac{dy}{dx})^2} \, dx\) beautifully captures this idea by integrating the square root of one plus the square of the derivative over the desired interval.

The skill of manipulating integrals, by transforming via substitutions—like using hyperbolic functions—is essential. It simplifies the work, especially when the integral is complex. This approach allows us to pair geometrical insights with mathematical transformations to solve problems that initially seem daunting, such as finding the arc length of a curved path.