Line integral

A line integral is the area under a function along a specified curve. An ordinary or Riemann integral is a line integral using the x-axis as the line and bound by a given interval on the x-axis.

A line integral is typically evaluated for the values taken by a three-dimensional vector field along a line or curve segment C. In this common use the line integral of a vector function f along C is the following (all three formulations are equivalent);

<math>\int_\mathbf{C} \vec{f}(\vec{s}) \cdot d\vec{s}</math>

<math>\int_\mathbf{C} f_1s_1 dx + f_2s_2 dy + f_3s_3 dz</math>

<math>\int_\mathbf{C} \langle \vec{f}(\vec{s}), d\vec{s} \rangle</math>