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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.50
Capitolo 11, Problema 11.PE.50

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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Identify the curves given: the cardioid is described by \(r = 2(1 + \sin \theta)\) and the circle by \(r = 2 \sin \theta\).
Determine the region of interest: the area inside the cardioid but outside the circle means we want the points where \(r\) lies between the circle and the cardioid, i.e., \(2 \sin \theta \leq r \leq 2(1 + \sin \theta)\).
Find the points of intersection by setting the two polar equations equal: \(2(1 + \sin \theta) = 2 \sin \theta\). Simplify this to find the values of \(\theta\) where the curves intersect.
Set up the integral for the area between the two curves using the formula for area in polar coordinates: \(A = \frac{1}{2} \int_{\alpha}^{\beta} \left( r_{outer}^2 - r_{inner}^2 \right) d\theta\), where \(r_{outer} = 2(1 + \sin \theta)\) and \(r_{inner} = 2 \sin \theta\).
Determine the limits of integration \(\alpha\) and \(\beta\) from the intersection points found earlier, then write the integral explicitly as \(A = \frac{1}{2} \int_{\alpha}^{\beta} \left[ (2(1 + \sin \theta))^2 - (2 \sin \theta)^2 \right] d\theta\).

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Polar Coordinates and Graphing

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how to graph curves like cardioids and circles in polar form is essential to visualize the regions described by equations such as r = 2(1 + sin θ) and r = 2 sin θ.
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Intro to Polar Coordinates

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) from θ = a to θ = b is given by the integral (1/2) ∫[a to b] (r(θ))^2 dθ. This formula is fundamental for finding areas bounded by one or more polar curves by setting appropriate limits and subtracting overlapping regions.
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Intro to Polar Coordinates

Determining Intersection Points and Integration Limits

To find the area between two polar curves, it is crucial to identify their points of intersection by solving r1(θ) = r2(θ). These intersection angles define the limits of integration, ensuring the correct portion of the region is calculated when subtracting the inner area from the outer.
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Determining Different Coordinates for the Same Point
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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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Identifying Conic Sections


Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.


x² + y² + 4x + 2y = 1

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Identifying Parametric Equations in the Plane


Exercises 1–6 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 and indicate the direction of motion and the portion traced by the particle.


x = √t, y = 1 − √t, t ≥ 0

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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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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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Polar to Cartesian Equations


Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.


r cos (θ − 3π/4) = (√2)/2

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