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→The Line Integral
=The Line Integral=
=The Line Integral=
==Definition==
We've integrated single variable functions along lines, and two variable functions over regions of the plane, and even three variable functions over regions of space. But lines, and curves, can be contained in the plane, and surfaces can be contained in space. Why have we been restricting ourselves to integrating functions over regions which have the same dimensions as their domains?
We've integrated single variable functions along lines, and two variable functions over regions of the plane, and even three variable functions over regions of space. But lines, and curves, can be contained in the plane, and surfaces can be contained in space. Why have we been restricting ourselves to integrating functions over regions which have the same dimensions as their domains?
<math>\int_C f(x,y)dL = \lim_{L_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k)L_k </math>
<math>\int_C f(x,y)dL = \lim_{L_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k)L_k </math>
==Computation==
As before, we're not going to be using the Riemann sum definition to compute line integrals. Let <math>x=x(t),y=y(t) </math> be a parametric description of <math>C </math>. This allows us to compute the length element, ''dL'':
<math>dL = \sqrt{dx^2+dy^2} </math>
But we don't have any formulas for ''dx'' or ''dy''! The closest we've got are formulas for <math>dx/dt, dy/dt</math>; we'll have to plug these in and see if we can patch things up:
<math>= \sqrt{ \left({ \frac{dx}{dt} }\right)^2 + \left({ \frac{dy}{dt} }\right)^2 } \sqrt{dt^2} </math>
Now we can just plug in our formula for ''dL'' into the integral to get an easy-to-use formula for a line integral:
<math>\int_C f(x,y)dL = \int_{t_0}^{t_1} f(x,y)\sqrt{ \left({ \frac{dx}{dt} }\right)^2 + \left({ \frac{dy}{dt} }\right)^2 } dt </math>
=Green's Theorem in the Plane=
=Green's Theorem in the Plane=