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 one leaf of r = cos 3θ
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16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
5:24 minutesProblem 12.3.47
Textbook Question
45–60. Areas of regions Find the area of the following regions.
The region inside the curve r = √(cos θ) and inside the circle r = 1/√2 in the first quadrant
Verified step by step guidance1
Identify the curves given: the polar curve is \( r = \sqrt{\cos \theta} \) and the circle is \( r = \frac{1}{\sqrt{2}} \). We are interested in the region inside both curves in the first quadrant, where \( \theta \) ranges from 0 to \( \frac{\pi}{2} \).
Find the points of intersection between the two curves by setting \( \sqrt{\cos \theta} = \frac{1}{\sqrt{2}} \). Square both sides to get \( \cos \theta = \frac{1}{2} \), then solve for \( \theta \) in the first quadrant.
Determine the limits of integration for \( \theta \) based on the intersection point and the first quadrant. The region inside both curves will be bounded by \( \theta = 0 \) and the intersection angle found in the previous step.
Set up the integral for the area of the region inside both curves. Since the region is inside both curves, the radius at each \( \theta \) is the smaller of the two radii. This means the area is the integral from 0 to the intersection angle of the area under \( r = \sqrt{\cos \theta} \), plus the integral from the intersection angle to \( \frac{\pi}{2} \) of the area under \( r = \frac{1}{\sqrt{2}} \). Use the polar area formula: \[ A = \frac{1}{2} \int_{\alpha}^{\beta} r(\theta)^2 \, d\theta \].
Write the final expression for the area as the sum of two integrals: \[ A = \frac{1}{2} \int_0^{\theta_0} \cos \theta \, d\theta + \frac{1}{2} \int_{\theta_0}^{\frac{\pi}{2}} \frac{1}{2} \, d\theta \], where \( \theta_0 \) is the intersection angle found earlier.
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Curves
Polar coordinates represent points using a radius and angle (r, θ). Curves defined in polar form, like r = √(cos θ), describe shapes based on radius as a function of angle. Understanding how to interpret and plot these curves is essential for setting up area integrals.
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Intro to Polar Coordinates
Area Calculation in Polar Coordinates
The area enclosed by a polar curve between angles θ = a and θ = b is given by (1/2) ∫ from a to b of r(θ)^2 dθ. This formula accounts for the sector-like shape of regions in polar form and is key to finding areas bounded by one or more polar curves.
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Determining Intersection Points and Integration Limits
To find the area common to two curves, it is necessary to find their points of intersection by equating their r-values and solving for θ. These intersection points define the limits of integration, ensuring the area is calculated only over the desired region, such as the first quadrant.
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