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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 11.5.18
Chapter 11, Problem 11.5.18

Finding Polar Areas


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


Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

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First, identify the curves given in polar coordinates: the circle is described by \(r = 4 \sin \theta\) and the line by \(r = 3 \csc \theta\). Understand that \(\csc \theta = \frac{1}{\sin \theta}\), so the line can be rewritten as \(r = \frac{3}{\sin \theta}\).
Next, determine the region of interest: the area inside the circle \(r = 4 \sin \theta\) and below the line \(r = 3 \csc \theta\). This means we want the set of points where \(r\) is less than or equal to \(4 \sin \theta\) and also less than or equal to \(3 \csc \theta\), with \(r\) values corresponding to points below the line.
Find the points of intersection between the two curves by setting \(4 \sin \theta = 3 \csc \theta\). Rewrite \(\csc \theta\) as \(\frac{1}{\sin \theta}\) to get \(4 \sin \theta = \frac{3}{\sin \theta}\). Multiply both sides by \(\sin \theta\) to obtain \(4 \sin^2 \theta = 3\). Solve for \(\sin \theta\) to find the values of \(\theta\) where the curves intersect.
Determine the limits of integration for \(\theta\) based on the intersection points and the region described (inside the circle and below the line). These limits will define the angular bounds for the area integral.
Set up the integral for the area in polar coordinates using the formula \(A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\). Since the region is bounded by two curves, express the area as the integral of the difference of the squares of the radii: \(A = \frac{1}{2} \int_{\alpha}^{\beta} \left( (4 \sin \theta)^2 - (3 \csc \theta)^2 \right) d\theta\). This integral will give the area of the region inside the circle and below the line.

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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 = 4 sin θ and r = 3 csc θ is essential to visualize the regions whose areas are to be found.
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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 found using the integral (1/2) ∫ from α to β of [r(θ)]² dθ. This formula is fundamental for computing areas bounded by one or more polar curves.
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Intro to Polar Coordinates

Determining Intersection Points and Limits of Integration

To find the area of the region inside one curve and below another, it is crucial to find their points of intersection by solving r-values and corresponding θ. These intersection points define the limits of integration for the area calculation.
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Determining Different Coordinates for the Same Point
Related Practice
Textbook Question

Implicitly Defined Parametrizations


Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.


x sin t + 2x = t, t sin t − 2t = y, t = π

Textbook Question

Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the line segment with endpoints (-1,3) and (3,-2)

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 = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

Textbook Question

Symmetries and Polar Graphs


Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.


r = 1 + 2 sin θ

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Textbook Question

Theory and Examples


Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

Textbook Question

Shifting Conic Sections


Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.


9x² + 6y² + 36y = 0