Arc Length
Length of a curve between two points.
Formula
Variables
Example
Measures curve distance
How it works
This formula computes the exact length of a smooth curve y = f(x) from x = a to x = b by adding up tiny straight-line pieces. Over a very small interval, the curve behaves almost like a right triangle with horizontal side dx and vertical side f'(x)dx, so each piece has length approximately sqrt((dx)^2 + (f'(x)dx)^2) = sqrt(1 + [f'(x)]^2) dx. Integrating these infinitesimal lengths gives the total arc length.
When to use it
Use this when you need the true distance along a graph, not just the horizontal distance between endpoints. It appears in calculus, physics, engineering, and geometry problems involving tracks, cables, road profiles, or any curve given as y = f(x).
Worked example, step by step
Find the arc length of y = (1/2)x^2 from x = 0 to x = 2. Step 1: Differentiate the function. f'(x) = x Step 2: Substitute into the arc length formula. L = ∫[0 to 2] sqrt(1 + [f'(x)]^2) dx = ∫[0 to 2] sqrt(1 + x^2) dx Step 3: Use the antiderivative of sqrt(1 + x^2). ∫ sqrt(1 + x^2) dx = (x/2)sqrt(1 + x^2) + (1/2)ln(x + sqrt(1 + x^2)) + C Step 4: Evaluate from 0 to 2. L = [(x/2)sqrt(1 + x^2) + (1/2)ln(x + sqrt(1 + x^2))] from 0 to 2 L = (2/2)sqrt(5) + (1/2)ln(2 + sqrt(5)) - 0 - (1/2)ln(1) L = sqrt(5) + (1/2)ln(2 + sqrt(5)) Final answer: L = sqrt(5) + (1/2)ln(2 + sqrt(5)) ≈ 2.958
Common mistakes to avoid
- Using the formula for a function that is not written as y = f(x) without first rearranging or choosing the correct variable
- Forgetting to square the derivative inside the square root, which changes the length completely
- Plugging in the endpoints before integrating instead of evaluating the antiderivative after the integral is found
Did You Know?
Arc-length integrals can be deceptively hard: even the simple parabola has no elementary arc-length formula.
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