High School Calculus/The Length of a Plane Curve

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Length of a Plane Curve[edit]

The graph of y = x^{\frac {3}{2}} 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 (\Delta x)^2 + (\Delta y)^2. Within that right triangle, the height \Delta y is the slope \left (\frac {\Delta y}{\Delta x}\right) times \Delta x. This secant slope is close to the slope of the curve. Thus \Delta y is approximately \left (\frac {\operatorname {d}y}{\operatorname {d}x}\right)\Delta x.

\Delta s \approx \sqrt{(\Delta x)^2 + \left (\frac {\operatorname {d}y}{\operatorname {d}x}\right)^2(\Delta x)^2} = \sqrt{1 + \left (\frac {\operatorname {d}y}{\operatorname {d}x}\right)^2}\Delta x (1)

Now add these pieces and make them smaller. The infinitesimal triangle has (\operatorname {d}s)^2 = (\operatorname {d}x)^2 + (\operatorname {d}y)^2. Think of \operatorname {d}s as \sqrt{1 + \left(\frac {\operatorname {d}y}{\operatorname {d}x}\right)^2}\operatorname {d}x and integrate:

length of curve = \int \operatorname {d}s = \int \sqrt {1 + \left(\frac {\operatorname {d}y}{\operatorname {d}x}\right)^2}\operatorname {d}x. (2)