Integral Arc Length Calculator

Calculate the arc length of a curve using integral calculus. This calculator computes the exact length of a curve defined by a function between two points using the arc length formula.

What is Arc Length?

Arc length is the distance along a curved path between two points. Unlike straight-line distance, arc length accounts for the curvature of the path. In calculus, we calculate arc length using definite integrals of the square root of the derivative of the function squared.

Arc length is important in physics, engineering, and geometry for measuring curved surfaces, designing bridges, and analyzing motion along curves.

Arc Length Formula

For a function y = f(x) between x = a and x = b, the arc length L is:

L = ∫_a^b √(1 + (dy/dx)²) dx

The formula integrates the square root of one plus the square of the derivative of the function. This accounts for the curvature of the curve.

Note: The function must be continuous and differentiable on the interval [a, b].

How to Use the Calculator

Enter the function y = f(x) in the input field. Use standard mathematical notation (e.g., "x^2" for x²).
Specify the lower limit (a) and upper limit (b) of the interval.
Click "Calculate" to compute the arc length.
The result will display the arc length in the same units as your x-axis.

The calculator uses numerical integration to approximate the arc length when an exact solution isn't possible.

Worked Example

Example Calculation

Find the arc length of y = x² from x = 0 to x = 1.

1. The derivative dy/dx = 2x.

2. The integrand is √(1 + (2x)²) = √(1 + 4x²).

3. The integral is ∫₀¹ √(1 + 4x²) dx.

4. The exact value is (1/8) [ (2x)√(1 + 4x²) + arcsinh(2x) ] from 0 to 1.

5. Numerical approximation gives approximately 1.1107.

This example shows how the calculator computes arc length for a simple quadratic function.

FAQ

What if the function is not continuous?: The arc length formula requires the function to be continuous and differentiable on the interval. If the function has breaks or sharp corners, the calculator may not work correctly.
Can I calculate arc length for parametric curves?: This calculator works for functions of the form y = f(x). For parametric curves (x = f(t), y = g(t)), you would need a different formula involving both derivatives.
How accurate are the results?: The calculator uses numerical integration with adaptive step size to provide accurate results. For simple functions, results are typically accurate to several decimal places.
What units should I use?: The arc length will be in the same units as your x-axis. For example, if x is in meters, the arc length will be in meters.
Can I calculate arc length for polar curves?: This calculator is designed for Cartesian coordinates. For polar curves (r = f(θ)), you would need a different formula involving the derivative of r with respect to θ.