Assume a function that maps points on a curve (yellow) to a function (blue) that uses the values of those curve points. Integration involves picking a starting point and an ending point on the curve and calculating the area between the function and the curve. For the above picture, the first point is on the x-axis and the last point is on the y-axis (the z-axis is used for the value of the function).
With this in mind we could say that the work you’ve done earlier integrating y = f(x) was a line integral and the curve was the straight line on the x-axis between x=a and x=b.
- blue — function f(x,y)
- yellow – segment C of curve S
We set up the integral by thinking about those rectangles. The height of each rectangle is F(x,y). The width of each rectangle is an infinitesimal piece of the yellow curve, and we will call that dS.
We will need to use math to get from dS to dx and dy.
Rob Sterling 