63–74. Arc length of polar curves Find the length of the following polar curves.


{Use of Tech} The complete limaçon r=4−2cosθ

Verified step by step guidance

1. Recall the formula for the arc length \( L \) of a polar curve \( r = r(\theta) \) from \( \theta = a \) to \( \theta = b \): \[ L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{d r}{d \theta}\right)^2} \, d\theta \]
2. Identify the given polar curve: \( r = 4 - 2 \cos \theta \). Since the problem asks for the complete limaçon, the interval for \( \theta \) is from \( 0 \) to \( 2\pi \).
3. Compute the derivative \( \frac{d r}{d \theta} \) with respect to \( \theta \): \[ \frac{d r}{d \theta} = \frac{d}{d \theta} (4 - 2 \cos \theta) = 2 \sin \theta \]
4. Substitute \( r(\theta) \) and \( \frac{d r}{d \theta} \) into the arc length formula: \[ L = \int_{0}^{2\pi} \sqrt{(4 - 2 \cos \theta)^2 + (2 \sin \theta)^2} \, d\theta \]
5. Simplify the expression inside the square root as much as possible before integrating, then evaluate the integral over \( 0 \leq \theta \leq 2\pi \) to find the total arc length.

Key Concepts

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 - 2cosθ. Understanding how to interpret and plot these curves is essential for analyzing their properties, including arc length.

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 combines the radius and its rate of change to measure the curve's length accurately. Applying this formula requires computing the derivative dr/dθ and evaluating the integral over the specified interval.

Limaçon Curves and Their Properties

A limaçon is a type of polar curve characterized by equations like r = a + b cosθ or r = a + b sinθ. Depending on parameters, it can have loops or dimpled shapes. Recognizing the limaçon form helps anticipate the curve's behavior and the interval for θ when calculating the complete arc length.