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
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16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
4:10 minutesProblem 71
Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The curve r = sin³(θ/3), for 0 ≤ θ ≤ π/2
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 function and interval: \( r(\theta) = \sin^3\left(\frac{\theta}{3}\right) \) and \( \theta \) ranges from 0 to \( \frac{\pi}{2} \).
Compute the derivative \( \frac{d r}{d \theta} \) using the chain rule:
First, write \( r(\theta) = \left( \sin\left(\frac{\theta}{3}\right) \right)^3 \).
Then,
\[
\frac{d r}{d \theta} = 3 \sin^2\left(\frac{\theta}{3}\right) \cdot \cos\left(\frac{\theta}{3}\right) \cdot \frac{1}{3} = \sin^2\left(\frac{\theta}{3}\right) \cos\left(\frac{\theta}{3}\right)
\]
Substitute \( r(\theta) \) and \( \frac{d r}{d \theta} \) into the arc length formula:
\[
L = \int_0^{\frac{\pi}{2}} \sqrt{ \sin^6\left(\frac{\theta}{3}\right) + \left( \sin^2\left(\frac{\theta}{3}\right) \cos\left(\frac{\theta}{3}\right) \right)^2 } \, d\theta
\]
Simplify the expression inside the square root if possible, then evaluate the integral over the interval \( 0 \leq \theta \leq \frac{\pi}{2} \) to find the arc length.
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Video duration:
4mKey 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 = sin³(θ/3). 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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Differentiation of Polar Functions
To apply the arc length formula, one must compute the derivative dr/dθ accurately. This involves differentiating functions like r = sin³(θ/3) using the chain rule and power rule, which is crucial for evaluating the integral that determines the curve's length.
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05:32Intro to Polar Coordinates
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