This worksheet shows a geometrical representation of the path integral for planar curves. [br]If [math]c:[a,b]\rightarrow \mathbb R^2[/math] is of class [math]C^1[/math] and the composite function [math]t\rightarrow f(x(t),y(t))[/math] is continuous, then we define[br][math]\quad \quad\quad\int_c fds=\int_a^bf(x(t),y(t))||c'(t)||dt[/math][br]When [math]f(x,y)\geq0[/math], this integral has a geometric interpretation as the [b]area of a fence[/b].

Move the points in the plane to change the shape of the curve. You can change the surface as well.