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 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: \( r = 4 + 4 \sin \theta \). We need to find \( \frac{d r}{d \theta} \), the derivative of \( r \) with respect to \( \theta \).
3. Compute the derivative: \[ \frac{d r}{d \theta} = 4 \cos \theta \]
4. 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 \( b = 2\pi \).
5. 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.

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 + 4 sin θ. 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 ∫ 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.

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.