Textbook Question
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 4 cos 2θ; at the tips of the leaves
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16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
3:34 minutesProblem 12.3.33
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 curve r = √(cos θ)
Verified step by step guidance1
First, understand the curve given in polar coordinates: \(r = \sqrt{\cos \theta}\). Since \(r\) must be real, identify the domain of \(\theta\) where \(\cos \theta \geq 0\). This will help determine the bounds for integration.
Sketch the curve by plotting points for values of \(\theta\) within the domain found. Note that \(r\) is non-negative and depends on \(\cos \theta\), which is positive in the intervals \([-\frac{\pi}{2}, \frac{\pi}{2}]\) and \([\frac{3\pi}{2}, \frac{5\pi}{2}]\), but since \(r\) is defined as \(\sqrt{\cos \theta}\), focus on the principal interval where \(r\) is real and positive.
Recall the formula for the area enclosed by a polar curve between angles \(\alpha\) and \(\beta\) is \(A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\). Substitute \(r^2 = \cos \theta\) into this formula.
Set up the integral for the area as \(A = \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos \theta \, d\theta\), using the interval where \(r\) is real and positive.
Evaluate the integral by integrating \(\cos \theta\) over the interval, then multiply by \(\frac{1}{2}\). This will give the area of the region inside the curve.
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Video duration:
3mKey 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 an angle (r, θ). Curves defined in polar form, like r = √(cos θ), describe shapes based on these parameters. Understanding how to interpret and plot such curves is essential for visualizing the region whose area is 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) from θ = a to θ = b is given by the integral (1/2) ∫[a to b] (r(θ))^2 dθ. This formula accounts for the sector-like slices of the region, making it crucial for finding areas bounded by polar curves.
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Determining the Limits of Integration
To compute the area, one must identify the correct interval [a, b] for θ where the curve is defined and the region is enclosed. This often involves analyzing where r(θ) is real and non-negative, and where the curve completes a full loop or returns to the starting point.
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