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 guidance

1. Recall the formula for the arc length \( L \) of a curve given in polar coordinates \( r = r(\theta) \) from \( \theta = a \) to \( \theta = b \): \[ L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{d}{d\theta}r(\theta)\right)^2} \, d\theta \]
2. Identify the given function and interval: here, \( r(\theta) = \theta^2 \) and \( \theta \) ranges from 0 to \( 2\pi \).
3. Compute the derivative of \( r(\theta) \) with respect to \( \theta \): \[ \frac{d}{d\theta}r(\theta) = \frac{d}{d\theta}(\theta^2) = 2\theta \]
4. Substitute \( r(\theta) \) and its derivative into the arc length formula to get the integrand: \[ \sqrt{(\theta^2)^2 + (2\theta)^2} = \sqrt{\theta^4 + 4\theta^2} \]
5. 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.

Key Concepts

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.

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.

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.