Arc Length Calculator
Compute the approximate arc length of a curve f(x) between two x-values using a constant average derivative. Use this for a quick linear approximation when the exact integral is complex.
About this calculator
The exact arc length of a smooth curve y = f(x) from x = a to x = b is given by L = ∫[a to b] √(1 + [f'(x)]²) dx. When f'(x) is approximated by a single constant average derivative value f'_avg, the integral simplifies to: L = (b - a) × √(1 + (f'_avg)²). This formula treats the curve as a straight line segment with the average slope, giving a reasonable estimate for nearly linear curves or as a first approximation. The term √(1 + (f'_avg)²) is the secant-line length scaling factor — it converts horizontal distance (b - a) into actual path length along the slope. More accurate results require splitting the interval into smaller subintervals and summing individual arc-length contributions.
How to use
Find the approximate arc length of a curve from x = 1 to x = 4 with an average derivative of f'_avg = 0.75. Here startPoint = 1, endPoint = 4, derivative = 0.75. Apply the formula: L = (4 - 1) × √(1 + 0.75²) = 3 × √(1 + 0.5625) = 3 × √1.5625 = 3 × 1.25 = 3.75. The approximate arc length is 3.75 units. A larger average derivative indicates a steeper curve and, therefore, a longer arc length relative to the horizontal distance.
Frequently asked questions
What is the arc length formula in calculus and how is it derived?
Arc length comes from summing infinitely many tiny straight-line segments along the curve. Each segment has horizontal length dx and vertical rise f'(x) dx, so its length is √(dx² + (f'(x) dx)²) = √(1 + [f'(x)]²) dx. Integrating over [a, b] gives the exact formula L = ∫[a to b] √(1 + [f'(x)]²) dx. This derivation uses the Pythagorean theorem applied to infinitesimally small right triangles beneath the curve.
Why does the arc length formula include the square root of 1 plus the derivative squared?
The expression √(1 + [f'(x)]²) converts a horizontal step of size dx into the actual slant distance along the curve at that point. When the slope f'(x) is zero (flat curve), the factor equals 1 and arc length equals horizontal distance. As the slope steepens, the factor grows, reflecting that the curve travels a longer path per unit of x. This is simply the Pythagorean theorem applied to a tiny right triangle with legs 1 and f'(x).
When is a constant average derivative a good approximation for arc length?
Using a single average derivative value works best when the curve's slope changes gradually and uniformly across the interval, meaning f''(x) is small. Over short intervals or nearly linear segments, the approximation can be quite accurate. For curves with sharp bends, steep changes, or high curvature, you should subdivide the interval and apply the formula piecewise, or use numerical integration of the full arc-length integral to avoid significant error.