Arc Length (Integral)
Length of a curve y = f(x) from a to b.
Formula
Variables
Example
For y = x from x=0 to 1: L = ∫√(1+1) dx = √2 ≈ 1.414.
How it works
This formula gives the exact length of a smooth curve y = f(x) between x = a and x = b. It comes from approximating the curve by many tiny straight-line segments; each tiny piece has length approximately √((dx)^2 + (dy)^2) = √(1 + (dy/dx)^2) dx, and adding all of them leads to the integral. The derivative f'(x) measures the local slope, so steeper parts contribute more to the total length.
When to use it
Use this when you need the true length of a graph segment rather than the horizontal distance, such as in physics, engineering, geometry, or curve design. It is especially useful when the curve is given as y = f(x) and you can differentiate it cleanly.
Worked example, step by step
Find the arc length of y = (2/3)x^(3/2) from x = 0 to x = 4. 1) Differentiate: f'(x) = d/dx[(2/3)x^(3/2)] = x^(1/2). 2) Substitute into the arc length formula: L = ∫[0 to 4] √(1 + (f'(x))^2) dx = ∫[0 to 4] √(1 + x) dx. 3) Integrate: ∫√(1+x) dx = ∫(1+x)^(1/2) dx = (2/3)(1+x)^(3/2). 4) Evaluate from 0 to 4: L = (2/3)(1+4)^(3/2) - (2/3)(1+0)^(3/2) = (2/3)(5^(3/2) - 1). 5) Numerical value: 5^(3/2) = 5√5 ≈ 11.1803, L ≈ (2/3)(11.1803 - 1) = (2/3)(10.1803) ≈ 6.7869. Final answer: L ≈ 6.787
Common mistakes to avoid
- using the function value f(x) instead of the slope f'(x) inside the square root
- forgetting to square the derivative term before adding 1
- mixing up the limits a and b or evaluating the antiderivative incorrectly
Did You Know?
Measuring the length of a curve requires integrating the Pythagorean theorem over infinitely many tiny steps.
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