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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.51
Chapter 12, Problem 12.3.51

45–60. Areas of regions Find the area of the following regions.


The region inside the rose r = 4 sin 2θ and inside the circle r = 2

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First, understand the problem: we need to find the area of the region that lies both inside the rose curve given by \(r = 4 \sin 2\theta\) and inside the circle given by \(r = 2\). This means we are looking for the overlapping area between these two curves.
Next, find the points of intersection between the two curves by setting their \(r\) values equal: \(4 \sin 2\theta = 2\). Solve for \(\theta\) to determine the limits of integration for the overlapping region.
Recall that the area enclosed by a polar curve \(r(\theta)\) between angles \(\alpha\) and \(\beta\) is given by the integral \(\frac{1}{2} \int_{\alpha}^{\beta} r(\theta)^2 \, d\theta\). We will use this formula to find the areas inside each curve.
Determine which curve is inside the other in the interval between the points of intersection. The overlapping area is the integral of the smaller radius squared over the appropriate interval. Set up the integral for the area of the region inside both curves accordingly.
Finally, evaluate the integral(s) over the determined limits to find the area of the overlapping region. Remember to consider symmetry if applicable to simplify the calculation.

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Key Concepts

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

Polar Coordinates and Graphs

Polar coordinates represent points using a radius and an angle, useful for curves like roses and circles. Understanding how to interpret and plot equations like r = 4 sin 2θ and r = 2 is essential to visualize the regions involved.
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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 (1/2) ∫[a to b] (r(θ))^2 dθ. This formula is fundamental for finding areas bounded by polar curves, including overlapping regions.
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Intro to Polar Coordinates

Finding Intersection Points of Polar Curves

To find the common region inside two polar curves, determine their points of intersection by equating r-values and solving for θ. These intersection points set the limits of integration for calculating the shared area.
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Slope of Polar Curves
Related Practice
Textbook Question

Second derivative Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that y″x=(fʹ(t)g″(t)−gʹ(t)f″(t))/(fʹ(t))³.  

Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r² = 4 sin θ  

Textbook Question

Circles in general Show that the polar equation

r² - 2r r₀ cos(θ - θ₀) = R² - r₀²

describes a circle of radius R whose center has polar coordinates (r₀, θ₀)

Textbook Question

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π. 


r = 3/(1 - cos θ)

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Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ

Textbook Question

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-1, -π/3)

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