Textbook Question
45–60. Areas of regions Find the area of the following regions.
The region common to the circles r = 2 sin θ and r = 1
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16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
6:01 minutesProblem 12.3.63
Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete circle r = a sin θ, where a > 0
Verified step by step guidance1
Recall the formula for the arc length \( L \) of a curve given in polar coordinates \( r = r(\theta) \) from \( \theta = \alpha \) to \( \theta = \beta \):
\[
L = \int_{\alpha}^{\beta} \sqrt{r(\theta)^2 + \left(\frac{d r}{d \theta}\right)^2} \, d\theta
\]
Identify the given polar curve: \( r = a \sin \theta \), where \( a > 0 \). Since this represents a complete circle, the parameter \( \theta \) will vary from \( 0 \) to \( \pi \) to trace the entire circle.
Compute the derivative of \( r \) with respect to \( \theta \):
\[
\frac{d r}{d \theta} = a \cos \theta
\]
Substitute \( r \) and \( \frac{d r}{d \theta} \) into the arc length formula:
\[
L = \int_0^{\pi} \sqrt{(a \sin \theta)^2 + (a \cos \theta)^2} \, d\theta
\]
Simplify the expression inside the square root before integrating.
Evaluate the integral over the interval \( 0 \leq \theta \leq \pi \) to find the total length of the curve, which corresponds to the circumference of the circle described by \( r = a \sin \theta \).
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Video duration:
6mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Polar Curves
Polar coordinates represent points in the plane using a radius and an angle (r, θ). Polar curves are defined by equations relating r and θ, such as r = a sin θ, which describes a circle. Understanding how to interpret and plot these curves is essential for analyzing their properties.
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Intro to Polar Coordinates
Arc Length Formula for Polar Curves
The arc length of a polar curve r(θ) from θ = α to θ = β is given by the integral ∫ from α to β of √[r(θ)² + (dr/dθ)²] dθ. This formula accounts for changes in both radius and angle, allowing calculation of the curve's length in polar form.
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06:29Arc Length of Parametric Curves
Differentiation of Polar Functions
To apply the arc length formula, one must compute the derivative dr/dθ of the polar function r(θ). This involves differentiating trigonometric functions like sin θ, which is crucial for evaluating the integral and finding the exact length of the curve.
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05:32Intro to Polar Coordinates
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