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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.2.47a
Chapter 11, Problem 11.2.47a

Cycloid


a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).

Verified step by step guidance
1
Identify the parametric equations of the cycloid: \(x = a(t - \sin t)\) and \(y = a(1 - \cos t)\), where \(t\) is the parameter.
Recall the formula for the arc length \(L\) of a curve defined parametrically by \(x = x(t)\) and \(y = y(t)\) over an interval \(t \in [\alpha, \beta]\): \(L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\).
Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\): \(\frac{dx}{dt} = a(1 - \cos t)\) \(\frac{dy}{dt} = a \sin t\).
Substitute these derivatives into the arc length integral and simplify the integrand: \(L = \int_{0}^{2\pi} \sqrt{a^2(1 - \cos t)^2 + a^2 \sin^2 t} \, dt\) Simplify inside the square root to express the integrand in a simpler form.
Evaluate the integral over one full arch of the cycloid, which corresponds to \(t\) going from \(0\) to \(2\pi\), to find the length of one arch.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For the cycloid, both x and y are given in terms of t, allowing us to analyze the curve's properties by differentiating with respect to t.
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Arc Length of Parametric Curves

The length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found by integrating the square root of (dx/dt)^2 + (dy/dt)^2 over that interval. This formula accounts for the combined rate of change in both x and y directions.
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Properties of the Cycloid

A cycloid is the path traced by a point on the rim of a rolling circle. One arch corresponds to t going from 0 to 2π. Understanding this interval helps set the limits for integration when calculating the length of one arch.
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Related Practice
Textbook Question

Polar Coordinates


Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.


a. (2, π/2)

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Textbook Question

Graphing Conic Sections


Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.


y² = −(8/3)x

Textbook Question

Finding Parametric Equations


Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².


a. once clockwise.


(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Textbook Question

Cartesian to Polar Equations


Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.


x² + y² + 5y = 0

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Textbook Question

Cartesian to Polar Coordinates


Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.


b. (-3,0)

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Textbook Question

Shifting Conic Sections


You may wish to review Section 1.2 before solving Exercises 39-56.


The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.

a. Find the center, foci, vertices, and asymptotes of the new hyperbola.