Line Integrals

If \(f\) is defined on a smooth curve \(C\) parametrically-defined as \(\big(x(t), y(t)\big)\) for \(a \leq t \leq b,\) then the line integral — or more specifically the line integral with respect to arclength — of \(f\) along \(C\) is defined to be \[ \int\limits_C f(x,y) \,\mathrm{d}\ell = \int\limits_a^b f(x,y) \sqrt{\bigg(\frac{\mathrm{d}x}{\mathrm{d}t}\bigg)^2 + \bigg(\frac{\mathrm{d}y}{\mathrm{d}t}\bigg)^2} \,\mathrm{d}t \] where \(\mathrm{d}\ell\) is a tiny change in arclength. If C is not smooth, but is piecewise smooth, then the integral of \(f\) over \(C\) is the sum of the integrals of \(f\) over each of the smooth components.

This generalized to curves in three-dimensions as one would expect.

If the curve \(C\) can be expressed as the graph of a function \(y = f(x),\) then we may express the same line integral with respect to \(x\) instead: \[ \int\limits_C f(x,y) \,\mathrm{d}x = \int\limits_a^b f(x,y) \,x'(t) \,\mathrm{d}t \] and similarly if the curve is \(x = f(y).\) When line integrals with respect to \(x\) and with respect to \(y\) occur together, it’s conventional to write \[ \int\limits_C P(x,y) \,\mathrm{d}x + \int\limits_C Q(x,y) \,\mathrm{d}y = \int\limits_C P(x,y) \,\mathrm{d}x + Q(x,y) \,\mathrm{d}y \]

It’s helpful to recall that a line segment with initial point \(\bm{v}_0\) and terminal point \(\bm{v}_1\) can be parameterized as \(\bm{r}(t) = (1-t)\bm{v}_0 + t\bm{v}_1\) for \(0 \leq t \leq 1.\)

A curve’s parameterization determines an orientation; i.e. the direction along which the curve is traversed. Given a curve \(C\) with implied orientation, we’ll let \(-C\) denote the same curve traversed in the opposite direction.