What is the formula for arc length in polar coordinates?

The formula for arc length in polar coordinates is given by L = ∫θ₁θ₂ √(r² + (dr/dθ)²) dθ.

To find the arc length in polar coordinates, we need to integrate the distance formula along the curve. The distance formula in polar coordinates is given by dθ = √(r² + (dr/dθ)²) dθ, where r is the radius and dr/dθ is the derivative of r with respect to θ. This formula represents the infinitesimal length of an arc in polar coordinates.

To find the total arc length, we need to integrate this formula over the range of θ from θ₁ to θ₂. Therefore, the formula for arc length in polar coordinates is given by L = ∫θ₁θ₂ √(r² + (dr/dθ)²) dθ.

Let's consider an example to illustrate this formula. Suppose we have a curve given by r = 2 + cos(θ) for 0 ≤ θ ≤ π. To find the arc length of this curve, we need to evaluate the integral L = ∫₀^π √(r² + (dr/dθ)²) dθ.

First, we need to find the derivative of r with respect to θ. We have dr/dθ = -sin(θ), so (dr/dθ)² = sin²(θ). Substituting this into the formula for arc length, we get L = ∫₀^π √((2 + cos(θ))² + sin²(θ)) dθ.

This integral is not easy to evaluate analytically, so we can use numerical methods to approximate the value of L. For example, we can use Simpson's rule to approximate the integral. Using a step size of h = π/6, we get L ≈ 6.283, which is the approximate arc length of the curve.