length of curve calculator - Aaron Graves, PhDude Replica

Arc Length Calculator

Compute the length of a curve using numerical integration (Simpson's Rule). Choose a curve type, enter expressions, and click calculate.

Curve Type

Function f(x)

Use x as the variable. Supported: sin, cos, tan, sqrt, abs, log, exp, pi, e, and operators + - * / ^

Start of interval (a)

End of interval (b)

x(t)

y(t)

Start of parameter (t₀)

End of parameter (t₁)

Subintervals (even number preferred)

Higher values usually increase accuracy but take longer.

What is the length of a curve?

The length of a curve (also called arc length) measures how far you would travel if you traced the curve from one endpoint to the other. Unlike straight-line distance, arc length follows every bend and slope change.

Arc length is used in physics, engineering, computer graphics, robotics, road design, and many other fields where curved paths matter.

Formulas used in this calculator

1) Cartesian form: y = f(x)

L = ∫ from a to b of √(1 + (dy/dx)²) dx

When your curve is expressed as y in terms of x, the calculator estimates the derivative dy/dx numerically and then integrates the expression above across your interval.

2) Parametric form: x = x(t), y = y(t)

L = ∫ from t₀ to t₁ of √((dx/dt)² + (dy/dt)²) dt

For parametric curves, the calculator estimates dx/dt and dy/dt, then integrates the speed magnitude along the parameter interval.

How to use this length of curve calculator

Choose the curve type (Cartesian or Parametric).
Enter your equation(s) with explicit multiplication, like 2*x (not 2x).
Enter start and end values for the interval.
Choose the number of subintervals (e.g., 1000).
Click Calculate Length to see your result.

Example calculations

Example A: y = x² on [0, 1]

Set mode to Cartesian, enter x^2, and interval 0 to 1. The calculator gives a numerical approximation of the exact arc length.

Example B: Unit circle

Set mode to Parametric with x(t) = cos(t), y(t) = sin(t), and t from 0 to 2*pi. The result is close to 2π, the circumference of a unit circle.

Tips for better accuracy

Increase subintervals for highly curved or oscillating functions.
Avoid discontinuities in the selected interval.
If values are very large/small, rescale the problem if possible.
Use smooth expressions and check for valid domains (e.g., no sqrt of negative values when unintended).

Common input mistakes

Using implicit multiplication (2x) instead of 2*x.
Mismatched parentheses.
Invalid interval values.
Typing unsupported symbols or text.

Final thoughts

This tool is designed for quick, practical arc-length estimates directly in your browser. It is ideal for learning calculus concepts, checking homework, prototyping engineering curves, or validating geometric models before moving to heavier software.