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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.1.21
Capitolo 12, Problema 12.1.21

15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.


x = cos t, y = sin² t; 0 ≤ t ≤ π

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Start with the given parametric equations: \(x = \cos t\) and \(y = \sin^{2} t\), where \(0 \leq t \leq \pi\).
Recall the Pythagorean identity: \(\sin^{2} t + \cos^{2} t = 1\). Use this to express \(\sin^{2} t\) in terms of \(\cos t\).
Since \(y = \sin^{2} t\), rewrite it as \(y = 1 - \cos^{2} t\). Substitute \(x = \cos t\) into this to eliminate the parameter \(t\).
This substitution gives the Cartesian equation relating \(x\) and \(y\): \(y = 1 - x^{2}\).
To describe the curve, recognize that \(y = 1 - x^{2}\) is a downward-opening parabola. The parameter range \(0 \leq t \leq \pi\) corresponds to \(x\) values from \(1\) to \(-1\). The positive orientation follows the direction of increasing \(t\), which moves from \(x=1\) to \(x=-1\) along the curve.

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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 and motions.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to find a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to remove t and express y explicitly or implicitly in terms of x.
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05:59
Eliminating the Parameter

Curve Orientation and Description

The orientation of a parametric curve indicates the direction in which the curve is traced as the parameter increases. Describing the curve involves identifying its shape (e.g., circle, ellipse) and specifying the interval of t to understand the portion and direction of the curve.
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Summary of Curve Sketching
Pratica correlata
Domanda del libro di testo

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = 3

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81–88. Arc length Find the arc length of the following curves on the given interval.


x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π

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13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


10x² - 7y² = 140

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33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside the circle r = 8 sin θ

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45–60. Areas of regions Find the area of the following regions.


The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ

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63–74. Arc length of polar curves Find the length of the following polar curves.


The complete circle r = a sin θ, where a > 0

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