arclength_surfarea - (9.1 Arc Length An interesting...

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(9.1) Arc Length An interesting geometric application of integration is the problem of finding the length of a curve. We imagine a curve (given by the graph of a function) as a fine wire, which we cut out and carefully straighten, and whose length we can measure. We call that length arc length. The recipe is as follows: Suppose the curve is y = f ( x ), for a x b . Then its arc length is L = b a 1 + [ f ( x )] 2 dx = b a 1 + dy dx 2 dx. Why does this formula work? Imagine a tiny piece of curve at x , of horizontal width dx . Call the length of that tiny piece of curve ds . (This is the differential of arc length.) Since the piece of curve is small, we pretend it is a little line segment. It forms the hypotenuse of a little right triangle, with base dx and the vertical leg dy . (Here, dy dx is the slope of the curve at that point.) The theorem of Pythagorus says ds 2 = dx 2 + dy 2 . So the length of that little piece of curve is ds = dx 2 + dy 2 = 1 + dy 2 dx 2 dx 2 = 1 + dy dx 2 dx.
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