Textbook Question
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the limaçon r = 2 + cos θ
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16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
3:06 minutesProblem 12.3.45
Textbook Question
45–60. Areas of regions Find the area of the following regions.
The region outside the circle r = 1/2 and inside the circle r = cos θ
Verified step by step guidance1
First, understand the problem: we want to find the area of the region that lies outside the circle given by \(r = \frac{1}{2}\) and inside the circle given by \(r = \cos \theta\) in polar coordinates.
Next, find the points of intersection between the two circles by setting their equations equal: \(\frac{1}{2} = \cos \theta\). Solve this equation for \(\theta\) to determine the limits of integration.
Recall that the area enclosed by a polar curve \(r = f(\theta)\) between angles \(\alpha\) and \(\beta\) is given by the integral \(\frac{1}{2} \int_{\alpha}^{\beta} (f(\theta))^2 \, d\theta\). We will use this formula to find the areas inside each circle.
Set up the integral for the area inside \(r = \cos \theta\) but outside \(r = \frac{1}{2}\) by subtracting the area inside \(r = \frac{1}{2}\) from the area inside \(r = \cos \theta\) over the interval between the intersection points found in step 2.
Finally, evaluate the integral expressions (or leave them in integral form if not asked to compute) to express the area of the desired region.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey 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 (r, θ). Understanding how to graph and interpret curves like r = 1/2 and r = cos θ is essential to visualize the regions bounded by these curves.
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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 of regions defined by polar curves.
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Determining Intersection Points and Limits of Integration
To find the area between two polar curves, it is crucial to identify their points of intersection by solving r1 = r2. These intersection angles set the limits for integration when calculating the enclosed area.
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6:02Determining Different Coordinates for the Same Point
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