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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.R.23
Chapter 12, Problem 12.R.23

22–23. Arc length Find the length of the following curves.
x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4

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Recall the formula for the arc length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\): \[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Find the derivatives of the parametric functions with respect to \(t\): Calculate \(\frac{dx}{dt}\) for \(x = \cos 2t\) and \(\frac{dy}{dt}\) for \(y = 2t - \sin 2t\).
Substitute the derivatives into the arc length formula under the square root: \[\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\]
Simplify the expression inside the square root as much as possible to make the integral easier to evaluate.
Set up the definite integral for the arc length from \(t = 0\) to \(t = \frac{\pi}{4}\): \[L = \int_0^{\frac{\pi}{4}} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\] Then, evaluate the integral to find the length of the curve.

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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 as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with these is essential for calculating properties like arc length.
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Parameterizing Equations

Arc Length Formula for Parametric Curves

The arc length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is given by the integral of the square root of (dx/dt)² + (dy/dt)² dt. This formula sums the infinitesimal distances along the curve, providing the total length between the parameter limits.
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Arc Length of Parametric Curves

Differentiation of Trigonometric Functions

Calculating dx/dt and dy/dt requires differentiating trigonometric functions like cosine and sine. Knowing the derivatives of these functions and applying the chain rule correctly is crucial to find the velocity components needed for the arc length integral.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

24–26. Sets in polar coordinates Sketch the following sets of points.


4 ≤ r² ≤ 9

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Textbook Question

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The segment of the curve x=y ³ +y+1 that starts at (1, 0) and ends at (11, 2).

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Textbook Question

Eliminate the parameter in the parametric equations x=1+sin t, y=3+2 sin t, for 0≤t≤π/2, and describe the curve, indicating its positive orientation. How does this curve differ from the curve x=1+sin t, y=3+2 sin t, for π/2≤t≤π?

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Textbook Question

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.


r = 3/(1 - 2 cos θ)

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A set of parametric equations for a given curve is always unique.

Textbook Question

42–43. Intersection points Find the intersection points of the following curves.

r= √(cos3t) and r= √(sin3t)

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