Calculating the Arc Length of a Curve Examples 1

# Calculating the Arc Length of a Curve Examples

We will now go through some examples of calculating the arc length of a curve using the following two formulas, the first for curves in the form $y = f(x)$ for a ≤ b, and the second for curves in the form $x = g(y)$ for c ≤ d.

(1)\begin{align} L = \int_a^b \sqrt{1 + \left ( \frac{dy}{dx} \right )^2} \: dx \quad \mathbf{or} \quad L = \int_a^b \sqrt{1 + [f'(x)]^2} \: dx \end{align}

(2)
\begin{align} L = \int_c^d \sqrt{1 + \left ( \frac{dx}{dy} \right )^2} \: dy \quad \mathbf{or} \quad L = \int_c^d \sqrt{1 + [g'(y)]^2} \: dy \end{align}