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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 11.3.3a
Capitolo 11, Problema 11.3.3a

Polar Coordinates


Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.


a. (2, π/2)

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Understand that a point in polar coordinates is given as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured from the positive x-axis.
For the point \((2, \frac{\pi}{2})\), plot it by moving 2 units from the origin in the direction of the angle \(\frac{\pi}{2}\) radians (which corresponds to 90 degrees, pointing straight up along the positive y-axis).
To find all polar coordinates for this point, recall that polar coordinates are not unique. You can add or subtract multiples of \(2\pi\) to the angle \(\theta\) without changing the point, so one set of coordinates is \((2, \frac{\pi}{2} + 2k\pi)\) for any integer \(k\).
Also, consider that if you take the negative radius \(-r\), you can add \(\pi\) to the angle to represent the same point. So another set of coordinates is \((-2, \frac{\pi}{2} + (2k+1)\pi)\) for any integer \(k\).
Summarize all possible coordinates as the union of these two sets: \((2, \frac{\pi}{2} + 2k\pi)\) and \((-2, \frac{\pi}{2} + (2k+1)\pi)\), where \(k\) is any integer.

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Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle. Each point is defined by (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. This system is useful for describing locations in circular or rotational contexts.
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Intro to Polar Coordinates

Plotting Points in Polar Coordinates

To plot a point given in polar coordinates, start at the origin, rotate counterclockwise by the angle θ, then move outward along that direction by the distance r. For example, (2, π/2) means move 2 units upward since π/2 radians corresponds to 90 degrees.
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Intro to Polar Coordinates

Multiple Representations of Polar Coordinates

A single point in the plane can have multiple polar coordinate representations due to angle periodicity and sign of r. Adding 2π to the angle or using a negative radius with an adjusted angle gives equivalent coordinates, so all such forms must be considered when finding all polar coordinates of a point.
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Intro to Polar Coordinates
Pratica correlata
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Cartesian to Polar Coordinates


Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.


c. (−1, √3)

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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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Theory and Examples


Volume Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x² + 4y² = 36 about the y−axis.

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Cartesian to Polar Coordinates


Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.


b. (-3,0)

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Shifting Conic Sections


You may wish to review Section 1.2 before solving Exercises 39-56.


The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.

a. Find the center, foci, vertices, and asymptotes of the new hyperbola.

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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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