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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.PE.33
Capitolo 11, Problema 11.PE.33

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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Start with the given Cartesian equation of the circle: \(x^{2} + y^{2} + 5y = 0\).
Rewrite the equation to complete the square for the \(y\) terms. Group \(y^{2} + 5y\) together: \(x^{2} + (y^{2} + 5y) = 0\).
Complete the square for \(y^{2} + 5y\) by adding and subtracting \(\left(\frac{5}{2}\right)^{2} = \frac{25}{4}\) inside the equation: \(x^{2} + \left(y^{2} + 5y + \frac{25}{4}\right) - \frac{25}{4} = 0\).
Rewrite the equation as \(x^{2} + \left(y + \frac{5}{2}\right)^{2} = \frac{25}{4}\), which represents a circle with center at \((0, -\frac{5}{2})\) and radius \(\frac{5}{2}\).
Convert to polar coordinates using \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\). Substitute these into the equation: \(\left(r \cos{\theta}\right)^{2} + \left(r \sin{\theta} + \frac{5}{2}\right)^{2} = \frac{25}{4}\). This is the polar form of the circle's equation.

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Conversion between Cartesian and Polar Coordinates

This concept involves expressing points from the Cartesian coordinate system (x, y) in terms of polar coordinates (r, θ), where x = r cos θ and y = r sin θ. Understanding this conversion is essential to rewrite Cartesian equations in polar form.
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Intro to Polar Coordinates

Equation of a Circle in Cartesian Coordinates

A circle in Cartesian form is typically expressed as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Recognizing and rewriting the given equation into this standard form helps identify the circle’s center and radius.
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Intro to Polar Coordinates

Polar Form of a Circle Equation

After converting Cartesian coordinates to polar, the circle’s equation can be expressed in terms of r and θ. This often involves substituting x = r cos θ and y = r sin θ and simplifying to isolate r, which helps in sketching and understanding the circle in polar coordinates.
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Convert Equations from Polar to Rectangular
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Length in Polar Coordinates


Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.


r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2

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Graphing Conic Sections


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


y² = −(8/3)x

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Lengths of Curves


Find the lengths of the curves in Exercises 13–19.


x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2

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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.)

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Area in Polar Coordinates


Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.


Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

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Cycloid


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

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