Arclengden

Arclength, also known as the curve length, is the distance along a curve. In calculus, arclength is typically calculated using an integral. For a curve defined parametrically by functions x(t) and y(t) for t in [a, b], the arclength L is given by the integral from a to b of the square root of (dx/dt)^2 + (dy/dt)^2 dt. If the curve is given by a function y = f(x) from x = a to x = b, the arclength formula simplifies to the integral from a to b of the square root of 1 + (dy/dx)^2 dx. This formula is derived by approximating the curve with many small line segments and summing their lengths. As the number of segments approaches infinity, the sum of the lengths of these segments converges to the true arclength of the curve. The concept of arclength is fundamental in various fields of mathematics and physics, including differential geometry, mechanics, and computer graphics. It allows for the measurement of distances along curved paths, which is crucial for understanding motion, shape, and spatial relationships.