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
6:07 minutesProblem 12.3.65
Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The spiral r = θ², for 0 ≤ θ ≤ 2π
Verified step by step guidance1
Recall the formula for the arc length \( L \) of a curve given in polar coordinates \( r = r(\theta) \) from \( \theta = a \) to \( \theta = b \):
\[
L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{d}{d\theta}r(\theta)\right)^2} \, d\theta
\]
Identify the given function and interval: here, \( r(\theta) = \theta^2 \) and \( \theta \) ranges from 0 to \( 2\pi \).
Compute the derivative of \( r(\theta) \) with respect to \( \theta \):
\[
\frac{d}{d\theta}r(\theta) = \frac{d}{d\theta}(\theta^2) = 2\theta
\]
Substitute \( r(\theta) \) and its derivative into the arc length formula to get the integrand:
\[
\sqrt{(\theta^2)^2 + (2\theta)^2} = \sqrt{\theta^4 + 4\theta^2}
\]
Set up the integral for the arc length:
\[
L = \int_0^{2\pi} \sqrt{\theta^4 + 4\theta^2} \, d\theta
\]
This integral can then be evaluated (using appropriate methods) to find the length of the spiral.
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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 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 = θ². Understanding how to interpret and plot these curves is essential for analyzing their properties, including 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 ∫ₐᵇ √[r(θ)² + (dr/dθ)²] dθ. This formula accounts for changes in both radius and angle, allowing calculation of the curve's length by integrating over the specified interval.
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06:29Arc Length of Parametric Curves
Differentiation of Polar Functions
To apply the arc length formula, you must compute the derivative dr/dθ of the polar function r(θ). This involves differentiating r = θ² with respect to θ, which yields dr/dθ = 2θ. Accurate differentiation is crucial for correctly evaluating the integral for arc length.
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05:32Intro to Polar Coordinates
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