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
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16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
7:00 minutesProblem 12.3.67
Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete cardioid r = 4 + 4 sin θ
Verified step by step guidance1
Recall the formula for the arc length \( L \) of a polar curve \( r = r(\theta) \) from \( \theta = a \) to \( \theta = b \):
\[
L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{d r}{d \theta}\right)^2} \, d\theta
\]
Identify the given polar curve: \( r = 4 + 4 \sin \theta \). We need to find \( \frac{d r}{d \theta} \), the derivative of \( r \) with respect to \( \theta \).
Compute the derivative:
\[
\frac{d r}{d \theta} = 4 \cos \theta
\]
Determine the interval for \( \theta \) that traces the complete cardioid. Since cardioids are typically traced once as \( \theta \) goes from \( 0 \) to \( 2\pi \), set the limits of integration as \( a = 0 \) and \( b = 2\pi \).
Set up the integral for the arc length:
\[
L = \int_{0}^{2\pi} \sqrt{(4 + 4 \sin \theta)^2 + (4 \cos \theta)^2} \, d\theta
\]
This integral can then be simplified and evaluated to find the length of the cardioid.
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Key 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 = 4 + 4 sin θ. Understanding how to interpret and plot these curves is essential for analyzing their properties, including arc length.
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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 = a to θ = b is given by the integral ∫ from a to b 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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Differentiation of Polar Functions
To apply the arc length formula, you must differentiate r(θ) with respect to θ. This involves using standard differentiation rules on trigonometric functions like sin θ. Accurate computation of dr/dθ is crucial for evaluating the integral correctly.
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