Arc Length Integrator

Math Calculus • Applications of Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

3. Arc Length Integrator

Computes curve length using \(L=\int_a^b\sqrt{1+(y'(x))^2}\,dx\) (explicit) or \(L=\int_a^b\sqrt{(x'(t))^2+(y'(t))^2}\,dt\) (parametric), with automatic derivatives and numerical evaluation when needed.

Inputs

ready

Curve type

Explicit uses variable x. Parametric uses variable t.

Output format Tolerance (numeric)

Adaptive Simpson tolerance. Smaller = more accurate, slower.

Function \(y=f(x)\)

Allowed: + − * / ^, parentheses, pi, e, sin/cos/tan/sec/csc/cot, exp, ln/log, sqrt, abs, min/max.

Length units (optional)

Displayed next to the final length.

Lower bound \(a\)

Supports: pi, e, + − * / ^ and parentheses.

Upper bound \(b\)

Example for ln(x): from 1 to e.

Trace position (for point/tangent)

Shows a moving point on the curve segment and a tangent direction.

Options

Show grid

Compare to straight line (chord)

Show steps

Show curve

Highlight segment \([a,b]\)

Show tangent at trace point

Presets

Click a preset to load and evaluate.

Graph

curve segment \([a,b]\) chord trace point

Drag to pan • wheel/pinch to zoom. The highlighted segment is the part whose arc length is computed.

x: —, y: — zoom(px/unit): —

Result

—

Enter a curve and bounds, then press Evaluate.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

3. Arc Length Integrator — Theory

Arc length measures how long a curve is between two parameter values. This tool supports explicit curves \(y=f(x)\) and parametric curves \((x(t),y(t))\), computing derivatives automatically.

1) Arc length for an explicit curve \(y=f(x)\)

Over \(x\in[a,b]\), the arc length is:

\[ L=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx. \]

\(\frac{dy}{dx}\) measures how steep the curve is.
If the curve is almost horizontal, \(\frac{dy}{dx}\approx 0\) and the integrand is close to 1.
The integrand \(\sqrt{1+(y')^2}\) is always \(\ge 1\), so the arc length is always \(\ge\) the horizontal distance \(|b-a|\).

2) Arc length for a parametric curve \((x(t),y(t))\)

If the curve is given by \(x=x(t)\), \(y=y(t)\) for \(t\in[a,b]\), then:

\[ L=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt. \]

This formula comes from adding up tiny “movement vectors” \((dx,dy)\).
It works even when the curve fails the vertical-line test.

3) Example (from the prompt)

Arc length of \(y=\ln(x)\) from \(x=1\) to \(x=e\):

\[ y'=\frac{1}{x},\qquad L=\int_1^e \sqrt{1+\left(\frac{1}{x}\right)^2}\,dx \approx 1.895. \]

The integral usually does not simplify to a “nice” closed form, so numerical methods are commonly used.

4) “Compare to straight line” (chord length)

The straight line distance between endpoints is:

\[ d=\sqrt{(\Delta x)^2+(\Delta y)^2}. \]

A key fact: arc length is always \(\ge\) chord length (the straight line is the shortest path).

5) What the graph shows

The curve is plotted in blue.
The portion between the chosen bounds is highlighted in green.
The dashed chord (optional) shows the straight-line distance between endpoints.
A trace point and tangent help visualize local direction (slope/velocity).

Troubleshooting tips

If you get “Non-finite sample values,” your function may be undefined on part of \([a,b]\) (e.g., \(\ln(x)\) for \(x\le 0\), division by zero, etc.).
If the curve looks empty, click Auto fit, zoom out, or pan — the segment might be off-screen.
Use the correct variable: explicit mode expects x, parametric mode expects t.

Frequently Asked Questions

What formula does the arc length integrator use for y = f(x)?

For an explicit curve, it uses L = integral from a to b of sqrt(1 + (dy/dx)^2) dx. The derivative dy/dx is computed automatically from your input function.

How do I compute arc length for a parametric curve with this tool?

Select the parametric mode and enter x = x(t) and y = y(t) with bounds a and b in terms of t. The tool uses L = integral from a to b of sqrt((dx/dt)^2 + (dy/dt)^2) dt.

Why is there a tolerance setting for arc length?

Many arc-length integrals do not simplify to an elementary closed form, so the value is often found numerically. A smaller tolerance targets higher accuracy but may take longer to compute.

What does “compare to straight line (chord)” mean?

The chord is the straight-line distance between the endpoints of the curve segment, d = sqrt((Delta x)^2 + (Delta y)^2). Arc length is always at least as large as this chord length.

Which variable should I use in explicit and parametric modes?

Explicit mode expects the variable x in y = f(x), while parametric mode expects the variable t in x(t) and y(t). Using the wrong variable can cause evaluation errors or an empty plot.