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Arc length of a parametric curve

Gives the length of a curve traced by a parametrized path c(t) for t from a to b, by integrating the speed of the path. It is used whenever the length of a smooth curve in space or the plane must be computed.

s = \int_a^b \left|\boldsymbol{c}'(t)\right| \, \mathrm{d}t

What the symbols mean

arc length of the curve (same unit as the coordinates)
start value of the parameter t
end value of the parameter t
parametrized path of the curve
derivative of the path with respect to t, the velocity vector
parameter of the curve

About this formula

  • FieldMathematics
  • TopicIntegral calculus
  • LevelUniversity
  • Also calledarc length, arclength, path length, length of a curve, curve length, length of arc
  • SourceWikidata Q670036

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