High School Calculus/The Length of a Plane Curve

Length of a Plane Curve
Edit

The graph of   is a curve in the x-y plane. How long is that curve? A definite integral needs endpoints and we specify x = 0 and x = 4. The first problem is to know what "length function" to integrate.

Here is the unofficial reasoning that gives the length of the curve. A straight piece has  . Within that right triangle, the height   is the slope   times  . This secant slope is close to the slope of the curve. Thus   is approximately  .

  (1)

Now add these pieces and make them smaller. The infinitesimal triangle has  . Think of   as   and integrate:

length of curve =  . (2)