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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.3.66
Capitolo 11, Problema 11.3.66

Cartesian to Polar Equations


Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
(x + 2)² + (y − 5)² = 16"

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1
Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \((x + 2)^2 + (y - 5)^2 = 16\) to rewrite it in terms of \(r\) and \(\theta\).
Expand the terms inside the parentheses: \((r \cos{\theta} + 2)^2 + (r \sin{\theta} - 5)^2 = 16\).
Use algebraic expansion to write the equation as \(r^2 \cos^2{\theta} + 4r \cos{\theta} + 4 + r^2 \sin^2{\theta} - 10r \sin{\theta} + 25 = 16\).
Combine like terms and use the Pythagorean identity \(\cos^2{\theta} + \sin^2{\theta} = 1\) to simplify the equation to a form involving \(r\) and \(\theta\) only.

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Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how these systems describe points differently is essential for converting equations between them.
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Intro to Polar Coordinates

Conversion Formulas Between Cartesian and Polar Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). These allow substitution of x and y in Cartesian equations with expressions in terms of r and θ, enabling the rewriting of Cartesian equations into polar form.
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Intro to Polar Coordinates

Equation of a Circle in Cartesian and Polar Forms

A circle centered at (h, k) with radius r satisfies (x - h)² + (y - k)² = r² in Cartesian form. To express this in polar coordinates, substitute x and y with r cos(θ) and r sin(θ), then simplify to find an equation involving r and θ.
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Convert Equations from Polar to Rectangular
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Ellipses and Eccentricity


Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.


Vertices: (±10,0)

Eccentricity: 0.24

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Surface Area


Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.


x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis

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Finding Polar Areas


Find the areas of the regions in Exercises 9–18.


Shared by the circles r = 1 and r = 2 sin θ

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Finding Cartesian from Parametric Equations


Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.


x=√(t+1), y=√t, t ≥ 0

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Hyperbolas and Eccentricity


In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.


y² − x² = 4

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Hyperbolas and Eccentricity


Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.


Eccentricity: 1.25

Foci: (0, ±5)

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