84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is
L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.

Verified step by step guidance

1. Recall the formula for the arc length of a curve given in Cartesian coordinates: if a curve is parameterized by \( x(t) \) and \( y(t) \) for \( t \) in \( [a, b] \), then the arc length \( L \) is \( L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \).
2. Express the polar curve \( r = f(\theta) \) in Cartesian coordinates using the relationships \( x = r \cos \theta = f(\theta) \cos \theta \) and \( y = r \sin \theta = f(\theta) \sin \theta \).
3. Compute the derivatives \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \) using the product rule: \( \frac{dx}{d\theta} = f'(\theta) \cos \theta - f(\theta) \sin \theta \) and \( \frac{dy}{d\theta} = f'(\theta) \sin \theta + f(\theta) \cos \theta \).
4. Substitute these derivatives into the arc length formula: \( L = \int_\alpha^\beta \sqrt{\left(f'(\theta) \cos \theta - f(\theta) \sin \theta\right)^2 + \left(f'(\theta) \sin \theta + f(\theta) \cos \theta\right)^2} \, d\theta \).
5. Simplify the expression inside the square root by expanding and combining like terms, using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), to show that it reduces to \( \sqrt{f'(\theta)^2 + f(\theta)^2} \), thus proving the formula \( L = \int_\alpha^\beta \sqrt{f(\theta)^2 + f'(\theta)^2} \, d\theta \).

Key Concepts

Polar Coordinates and Curves

Polar coordinates represent points in the plane using a radius and an angle (r, θ). A polar curve is defined by a function r = f(θ), describing how the radius changes with the angle. Understanding this system is essential to relate the curve's shape to its parametric form.

Arc Length Formula for Parametric Curves

The arc length of a curve defined parametrically by x(t) and y(t) from t = a to b is given by the integral of the square root of (dx/dt)² + (dy/dt)² dt. This formula generalizes the distance traveled along a curve and is the foundation for deriving arc length in polar form.

Differentiation and Chain Rule in Polar Coordinates

To find the arc length of a polar curve, one must differentiate r = f(θ) and convert the curve into parametric form x = r cos θ, y = r sin θ. Applying the chain rule to these expressions allows computation of dx/dθ and dy/dθ, which are used in the arc length integral.