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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.5.10
Chapter 11, Problem 11.5.10

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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1
Identify the curves given: the circles are described by the polar equations \(r = 1\) and \(r = 2 \sin \theta\).
Find the points of intersection by setting the two equations equal: \(1 = 2 \sin \theta\). Solve for \(\theta\) to determine the limits of integration.
Determine which curve lies inside and which lies outside between the points of intersection by testing values of \(\theta\) in the interval.
Set up the integral(s) for the area of the region(s) bounded by the curves. Recall that the area enclosed by a polar curve \(r(\theta)\) from \(\alpha\) to \(\beta\) is given by \(\frac{1}{2} \int_{\alpha}^{\beta} r(\theta)^2 \, d\theta\).
Express the total area as the sum or difference of integrals of \(\frac{1}{2} r^2\) for each curve over the appropriate intervals, based on which curve is outer or inner in each region.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates and Graphing

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian coordinates. Understanding how to graph polar equations like r = 1 and r = 2 sin θ is essential to visualize the regions whose areas are to be found.
Recommended video:
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Intro to Polar Coordinates

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) between angles α and β is given by the integral (1/2) ∫ from α to β of [r(θ)]² dθ. This formula is fundamental for finding areas bounded by one or more polar curves.
Recommended video:
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Intro to Polar Coordinates

Finding Points of Intersection in Polar Curves

To determine the limits of integration, it is necessary to find where the polar curves intersect by solving r₁(θ) = r₂(θ). These intersection points define the boundaries of the shared region for accurate area calculation.
Recommended video:
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Slope of Polar Curves
Related Practice
Textbook Question

Identifying Graphs


Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x


Then find each parabola's focus and directrix.



Textbook Question

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

Textbook Question

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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Textbook Question

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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Textbook Question

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

Textbook Question

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