Problem 30 - Thomas Calculus

Question

Find the length of one arch of the cycloid \(x=a(\theta-\sin \theta)\) \(y=a(1-\cos \theta), 0 \leq \theta \leq 2 \pi,\) shown in the accompanying figure. A cycloid is the curve traced out by a point \(P\) on the circumference of a circle rolling along a straight line, such as the \(x\) -axis.

Step-by-Step Solution

Answer

The length of one arch of the cycloid is \(8a\).

Step 1: Formula for Arc Length

The arc length of a curve given by parametric equations \(x = f(t)\) and \(y = g(t)\) from \(t = a\) to \(t = b\) is calculated using the formula: \(S = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\).

Step 2: Differentiate x with Respect to θ

Given \(x = a(\theta - \sin \theta)\), differentiate with respect to \(\theta\):\[\frac{dx}{d\theta} = a(1 - \cos \theta)\]

Step 3: Differentiate y with Respect to θ

Given \(y = a(1 - \cos \theta)\), differentiate this with respect to \(\theta\):\[\frac{dy}{d\theta} = a\sin \theta\]

Step 4: Substitute into Arc Length Formula

Substitute \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) into the arc length formula:\[S = \int_0^{2\pi} \sqrt{(a(1 - \cos \theta))^2 + (a\sin \theta)^2} \, d\theta\]

Step 5: Simplify the Expression Under the Integral

Simplify the expression under the square root:\((a(1 - \cos \theta))^2 + (a\sin \theta)^2 = a^2((1 - \cos \theta)^2 + \sin^2 \theta)\)Expanding and combining like terms, we have:\[= a^2(1 - 2\cos \theta + \cos^2 \theta + \sin^2 \theta) = a^2(2 - 2\cos \theta) = 2a^2(1 - \cos \theta)\]

Step 6: Solve the Integral

Taking the square root of the simplified expression:\[ \sqrt{2a^2(1 - \cos \theta)} = a\sqrt{2(1 - \cos \theta)} = a\sqrt{4\sin^2(\theta/2)} = 2a\sin(\theta/2)\]The integral then becomes:\[ S = \int_0^{2\pi} 2a\sin(\theta/2) \, d\theta \]

Step 7: Evaluate the Integral

Now, evaluate the integral using the substitution method with \( u = \theta/2 \), which gives \(du = 1/2 \, d\theta\):\[ S = 4a \int_0^{\pi} \sin u \, du \]This integral evaluates to:\[ S = 4a [-\cos u]_0^{\pi} = 4a [1 - (-1)] = 8a \]

Step 8: Final Step: Conclusion

The length of one arch of the cycloid, from \(\theta = 0\) to \(\theta = 2\pi\), is therefore \(8a\).

Key Concepts

Arc Length

In mathematics, the arc length of a curve is the distance along the curve between two points. Finding the arc length is a crucial part of understanding the nature of complex curves like the cycloid. The formula used for calculating the arc length when dealing with parametric equations is:

• \( S = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \)

This formula involves integrating the square root of the sum of the squares of the derivatives of the curve's parametric equations. The limits of the integral \(a\) and \(b\) define the interval over which the arc length is measured, such as from \(t = 0\) to \(t = 2\pi\). Understanding this formula is fundamental to determining arc lengths in more complex curves.

Parametric Equations

Parametric equations are a way of defining a curve by representing its coordinates as functions of a single parameter. In the case of a cycloid, the parametric forms are given as:

• \( x = a(\theta - \sin \theta) \)
• \( y = a(1 - \cos \theta) \)

Here, \(\theta\) is the parameter which represents the angle of rotation as the circle rolls along the x-axis. Parametric equations are particularly useful because they can describe complex curves which cannot be easily expressed using the standard Cartesian coordinates. Understanding how to utilize parametric equations is essential for analyzing curves like the cycloid.

Integration

Integration is a mathematical technique used to calculate areas, volumes, and other quantities that add up across a curve or surface. In the context of finding the cycloid's arc length, integration helps sum the infinitesimal segments of the curve to find the total distance travelled by a point on the circle's rims. In this solution:

• The integral setup for the arc length is: \( S = \int_0^{2\pi} 2a\sin(\theta/2) \, d\theta \)

An integral can often be evaluated using substitution, which simplifies the function within the integral. By substituting \( u = \theta/2 \) and finding the corresponding differential, the integral can be solved to get the arc length of one arch of the cycloid. Grasping integration techniques is critical for solving such problems effectively.

Differentiation

Differentiation is the process of finding the derivative of a function. It determines the rate at which a function's value changes. In the context of parametric equations, differentiation with respect to the parameter provides insight into the curve's behavior over that parameter. For the cycloid:

• The derivatives are \(\frac{dx}{d\theta} = a(1 - \cos \theta)\) and \(\frac{dy}{d\theta} = a\sin \theta\).

These derivatives represent the rates of change of the x and y positions concerning the angle \(\theta\). Calculating these derivatives is crucial as they are used in the arc length formula to express how fast the curve is changing. A strong foundation in differentiation will help simplify the process of analyzing and solving for the arc length.