21
$\begingroup$

A line integral (with respect to arc length) can be interpreted geometrically as the area under $f(x,y)$ along $C$ as in the picture. You sum up the areas of all the infinitesimally small 'rectangles' formed by $f(x,y)$ and $ds$.

enter image description here

What I'm wondering is how do I interpret line integrals with respect to $x$ or $y$ geometrically?

  • 0
    You rather integrate with respect to the arclength, that's why you use $dS$. In turn, this will depend on $dx$ and $dy$ for cartesian cordinates or on $d\theta$ and $dr$ for polar coordinates, i.e. (and don't take this strictly) $$ds^2 = dx^2+dy^2$$ $$ds^2 = dr^2+r^2d\theta^2$$2012-03-04
  • 0
    You get the "algebraic sum" of the projected areas. E.g., if $C$ is a circle in the $(x,y)$-plane you get zero.2012-03-04

5 Answers 5