The Arc Length of Curve Calculator computes the length of any smooth polynomial curve y = ax² + bx + c over an interval using numerical integration. Extends the simple circular arc formula to arbitrary curves — used in calculus, computer graphics, differential geometry, and engineering.
4.646784
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4.472136
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1.039052
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4.646784
units
4.472136
units
1.039052
0
4
A straight line has length = distance. A circle has arc length = rθ. But what about the length of a parabola, a roller coaster track, or the path traced by a projectile? These require arc length integration — a calculus technique that adds up infinitely many tiny straight-line segments along the curve. The calculator for arc length of a curve computes this integral numerically for polynomial curves, giving the true path length without manual calculus.
For a smooth curve y = f(x) over the interval [x₁, x₂], the arc length L is:
L = ∫[x₁ to x₂] √[1 + (dy/dx)²] dx = ∫[x₁ to x₂] √[1 + (f′(x))²] dx
The integrand √[1 + (f′(x))²] is the length of an infinitesimal tangent element — derived from Pythagoras' theorem applied to infinitesimal changes dx and dy = f′(x)dx: ds = √(dx² + dy²) = √[1 + (dy/dx)²] dx. For a quadratic y = ax² + bx + c: f′(x) = 2ax + b, so the integrand becomes √[1 + (2ax + b)²], which has a known but complex analytical form. Numerical integration (Gaussian quadrature or Simpson's rule) efficiently computes this for any coefficients and interval. The arc length calculator handles the simpler circular arc case analytically.
The arc length always exceeds or equals the straight-line chord length between the two endpoints. The ratio arc/chord quantifies the "curviness" of the path:
A highway cloverleaf loop may have an arc/chord ratio of 2–3. A vertical circular loop in a rollercoaster has arc length πd (half-circumference) compared to chord length d — a ratio of π/2 ≈ 1.571. Understanding this ratio helps engineers calculate material quantities, travel times, and structural loads along curved paths.
Arc length of curves appears across technical disciplines:
The integral calculator and integrals calculators category provide the broader toolkit for calculus-based calculations.
For curves defined parametrically as x(t), y(t) over interval [t₁, t₂]: L = ∫√[(dx/dt)² + (dy/dt)²] dt. For polar curves r(θ) over [θ₁, θ₂]: L = ∫√[r² + (dr/dθ)²] dθ. These generalizations handle curves that cannot be expressed as single-valued functions of x — spirals, loops, and closed curves. The fundamental structure √(sum of squared derivatives) appears in all three forms, reflecting the underlying Pythagorean geometry of infinitesimal arc elements in the respective coordinate systems.
For the curve $$y = ax^2 + bx + c$$ over $$[x_1, x_2]$$, the arc length is:
$$L = \int_{x_1}^{x_2} \sqrt{1 + (2ax + b)^2}\,dx$$
This integral is evaluated numerically using composite Simpson's rule with $$n = 100$$ subintervals:
$$h = \frac{x_2 - x_1}{100}$$
$$L \approx \sum_{k=0}^{9} S_k$$
where each $$S_k$$ covers 10 subintervals using the Simpson pattern:
$$S_k = \frac{h}{3}\left[g_{10k} + 4g_{10k+1} + 2g_{10k+2} + \cdots + 4g_{10k+9} + g_{10k+10}\right]$$
and $$g_i = \sqrt{1 + (2a \cdot x_i + b)^2}$$ at each sample point $$x_i = x_1 + ih$$.
The chord length is the Euclidean distance between endpoints:
$$\text{Chord} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
The arc/chord ratio is $$L / \text{Chord}$$, which equals 1 for a straight line and exceeds 1 for any curved path.
The Arc Length is the total distance measured along the curve itself. If you imagined laying a flexible tape measure along the parabola, this is the reading you would get. It is always greater than or equal to the chord length.
The Chord Length is the straight-line (Euclidean) distance between the two endpoints of the curve. This represents the shortest possible path between the endpoints.
The Arc/Chord Ratio quantifies how much longer the curved path is compared to the straight line. A ratio of 1.0 means the curve is perfectly straight on the interval. Ratios above 1.5 indicate substantial curvature. This metric is used in road design (curves with high ratios require speed reductions) and in manufacturing (to estimate material waste along curved cuts).
The y values at the bounds confirm the function values at the endpoints, helping you visualize the curve's position.
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The parabola y = x^2 rises from (0,0) to (2,4). The derivative is f'(x) = 2x, so the integrand is sqrt(1+4x^2). The chord length is sqrt(4+16) = sqrt(20) = 4.4721. The arc length 4.6468 is about 3.9% longer than the chord, indicating mild curvature over this interval.
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This parabola has vertex at (2, -1) and returns to y=3 at both endpoints. The chord from (0,3) to (4,3) is a horizontal line of length 4. The arc dips down and back up, traveling 9.29 units — over 2.3 times the chord length. The high ratio reflects the U-shaped path.
For a general quadratic, the integrand $$\sqrt{1+(2ax+b)^2}$$ involves the square root of a polynomial. While this particular form can be expressed using inverse hyperbolic sine ($$\sinh^{-1}$$) or logarithms, the resulting expression is cumbersome. Numerical integration provides the same accuracy with simpler implementation.
Simpson's rule with 100 subintervals (effectively a composite rule of order 4) provides accuracy to approximately 8-10 significant digits for smooth integrands like $$\sqrt{1+(2ax+b)^2}$$. For the typical coefficient ranges in this calculator, the error is well below $$10^{-6}$$.
The ratio measures curvature severity. A ratio of 1.0 means the path is straight. Ratios of 1.01-1.05 indicate mild curvature. Ratios above 1.5 indicate significant bending. In road engineering, this ratio helps determine whether a curve requires speed advisories or banking.
Yes. Set $$a = 0$$ to get a linear function $$y = bx + c$$. The derivative is constant $$f'(x) = b$$, so the integrand is $$\sqrt{1+b^2}$$, and the arc length equals $$(x_2 - x_1)\sqrt{1+b^2}$$, which matches the Euclidean distance formula. The arc/chord ratio will be exactly 1.
Yes. The arc length formula involves $$(f'(x))^2$$, which is always non-negative regardless of whether the parabola opens upward ($$a > 0$$) or downward ($$a < 0$$). The arc length depends only on the shape of the curve, not its orientation.
Arc length and curvature are related but distinct concepts. Curvature $$\kappa$$ measures how sharply a curve bends at each point, while arc length measures total distance along the curve. For a parabola $$y = ax^2+bx+c$$, the curvature is $$\kappa = \frac{|2a|}{(1+(2ax+b)^2)^{3/2}}$$. Higher average curvature generally leads to a larger arc/chord ratio.
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