Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problema 12.3.1

Capitolo 12, Problema 12.3.1

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

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Recall the relationship between polar and Cartesian coordinates: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r\) is the radius and \(\theta\) is the angle parameter.

Given the polar equation \(r = f(\theta)\), substitute \(r\) into the Cartesian coordinate formulas to express \(x\) and \(y\) in terms of \(\theta\).

Write the parametric equations as \(x(\theta) = f(\theta) \cdot \cos(\theta)\) and \(y(\theta) = f(\theta) \cdot \sin(\theta)\).

Note that \(\theta\) serves as the parameter that varies, typically within an interval such as \([0, 2\pi]\), to trace the curve in the Cartesian plane.

Thus, the polar equation \(r = f(\theta)\) is expressed in parametric form as \(\left(x(\theta), y(\theta)\right) = \left(f(\theta) \cos(\theta), f(\theta) \sin(\theta)\right)\).

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Polar Coordinates

Polar coordinates represent points in a plane using a radius and an angle (r, θ) from the origin. The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential to convert between polar and Cartesian forms.

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Intro to Polar Coordinates

Parametric Equations

Parametric equations express coordinates as functions of a parameter, often denoted t or θ. Instead of y as a function of x, both x and y are defined separately in terms of the parameter, allowing more flexible representations of curves, including those defined in polar form.

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Parameterizing Equations

Conversion from Polar to Cartesian Coordinates

To convert polar equations to Cartesian form, use the relationships x = r cos(θ) and y = r sin(θ). When r is given as a function of θ, substituting r = f(θ) into these formulas yields parametric equations x(θ) and y(θ) in Cartesian coordinates.

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Intro to Polar Coordinates

Pratica correlata

Domanda del libro di testo

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

The region common to the circles r = 2 sin θ and r = 1

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31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(1, √3)

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37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The upper half of the parabola x=y ², originating at (0, 0)

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

The region inside the lemniscate r² = 6 sin 2θ

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Domanda del libro di testo

75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

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57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.

r = 2 - 2 sin θ b

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