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Chapter 6: Problem 30

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.

Short Answer

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

Step by step solution

01

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: \(S = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\).
02

Differentiate x with Respect to 胃

Given \(x = a(\theta - \sin \theta)\), differentiate with respect to \(\theta\):\[\frac{dx}{d\theta} = a(1 - \cos \theta)\]
03

Differentiate y with Respect to 胃

Given \(y = a(1 - \cos \theta)\), differentiate this with respect to \(\theta\):\[\frac{dy}{d\theta} = a\sin \theta\]
04

Substitute into Arc Length Formula

Substitute \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) into the arc length formula:\[S = \int_0^{2\pi} \sqrt{(a(1 - \cos \theta))^2 + (a\sin \theta)^2} \, d\theta\]
05

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:\[= a^2(1 - 2\cos \theta + \cos^2 \theta + \sin^2 \theta) = a^2(2 - 2\cos \theta) = 2a^2(1 - \cos \theta)\]
06

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:\[ S = \int_0^{2\pi} 2a\sin(\theta/2) \, d\theta \]
07

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:\[ S = 4a [-\cos u]_0^{\pi} = 4a [1 - (-1)] = 8a \]
08

Final Step: Conclusion

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. The region bounded by \(y=\sqrt{x}, y=2, x=0\) about a. the \(x\) -axis b. the \(y\) -axis c. the line \(x=4\) d. the line \(y=2\)

Designing a wok You are designing a wok frying pan that will be shaped like a spherical bowl with handles. A bit of experimentation at home persuades you that you can get one that holds about 3 \(\mathrm{L}\) if you make it 9 \(\mathrm{cm}\) deep and give the sphere a radius of \(16 \mathrm{cm} .\) To be sure, you picture the wok as a solid of revolution, as shown here, and calculate its volume with an integral. To the nearest cubic centimeter, what volume do you really get? \(\left(1 \mathrm{L}=1000 \mathrm{cm}^{3} .\right)\)

Find the surface area of the cone frustum generated by revolving the line segment \(y=(x / 2)+(1 / 2), 1 \leq x \leq 3,\) about the \(y-\) axis. Check your result with the geometry formula Frustum surface area \(=\pi\left(r_{1}+r_{2}\right) \times\) slant height.

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-22\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. \(x=y^{3} / 3, \quad 0 \leq y \leq 1 ; \quad y\) -axis

The region in the first quadrant that is bounded above by the curve \(y=1 / \sqrt{x},\) on the left by the line \(x=1 / 4,\) and below by the line \(y=1\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid by a. the washer method. b. the shell method.
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