![]() It follows that the width of the surface element is $r\ d\theta$ (the angle shrinks to an infinitesimal, but the radial coordinate does not - it simply takes the value of the radius wherever we place our surface element). It should be easy to convince yourself that the length of an arc that spans angle $theta$ is $r\theta$ (of course $theta$ in radians). But the length of the arc between two evenly spaced blue lines clearly increases as the radial coordinate increases. We don't need to worry about this because in the infinitesimal limit, they approach the same length. First, you might notice that the "inner width" and "outer width" are different. The length is easy, it is $dr$, and is always the same (notice that the length of a blue segment between the evenly spaced red lines is always the same). ![]() In the infinitesimal limit the area of one such segment is just its length multiplied by its width. ![]() The surface element has the same shape as one of the spaces between two red and two blue lines (a sort of curved rectangle). ![]() The reason for this can be seen geometrically: CC-BY-SA-3.0 Creative Commons Attribution-Share Alike 3.When integrating in (2D) polar coordinates you need to use a surface element: This licensing tag was added to this file as part of the GFDL licensing update.
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