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Arc Length Calculator

Measure the exact length of any curve between two points

The arc length of a curve y = f(x) from x = a to x = b is:

L = \int_a^b \sqrt{1 + [f'(x)]^2}\, dx

For parametric curves (x(t), y(t)):

L = \int_{t_1}^{t_2} \sqrt{[x'(t)]^2 + [y'(t)]^2}\, dt

This arc length calculator with graph highlights the curve segment on an interactive graph — see exactly which part of the curve is being measured. Type any function and a range, and the AI computes the exact arc length using numerical integration. Zoom, pan, and explore the curve visually. Works with polynomials, trigonometric, exponential, and parametric curves.

Graph

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FAQ

What is arc length?
Arc length is the distance along a curve between two points — if you "straightened out" the curve, that would be its arc length. It is always greater than or equal to the straight-line distance between the endpoints.
How is arc length calculated?
The arc length formula integrates \sqrt{1 + [f'(x)]^2} from a to b. Intuitively, at each tiny segment dx, the curve rises by f'(x)·dx, so the segment length is \sqrt{dx^2 + dy^2}. Summing these gives the total arc length.
Can I compute arc length of a parametric curve?
Yes. Type a parametric curve like "(cos(t), sin(t))" and a range like "t from 0 to 2pi". The formula uses \sqrt{(dx/dt)^2 + (dy/dt)^2} instead.
What if no bounds are given?
Arc length requires bounds — without them, the length is infinite for any non-constant function. Specify "from a to b" or the calculator uses the current viewport range.
How accurate is the calculation?
The calculator uses Simpson's rule with 500 subintervals, giving accuracy to about 6 decimal places for smooth functions. This is more than sufficient for all practical and educational purposes.
Why use an arc length calculator with a graph?
The graph shows exactly which part of the curve is being measured. You can see the curve, the endpoints, and understand visually why a curve is longer than the straight-line distance between its endpoints. Interactive zoom and pan let you explore the curve's shape in detail.

Related Topics

Integrals (lesson)Area Between Curves CalculatorTangent Line Calculator