PathLength
Author: Lenore Horner

This applet illustrates the use of differentiation and integration to calculate distance along a curved path.  See below for a few instructions about manipulating this.

Drag the red points A and B to change the limits of integration - the horizontal coordinates of the starting and ending point of the path.

On Mac, two-fingered-click on the blue function f(x) and choose the bottom option to redefine the path you measure the distance along.  Otherwise, try right-click on the function.

Under move, there is a zoom control.  In addition, on Macs, shift-drag moves the window, shift-drag on an axis rescales just that axis.

g(x)=Derivative[f(x)]

h(x)=sqrt(1+(g(x))^2)

f(x) = 0.7 * (x - 0.5) * (x - 2.0) * (x - 4.0) * (x - 5.25)
g = Derivative(f(x))
h(x) = sqrt(1.0 + g(x)^(2.0))
A = Point(xAxis)
B = Point(xAxis)
C: (x(A), f(x(A)))
D: (x(B), f(x(B)))
a = Segment(C, D)
b = Segment(A, B)
DistanceABviaf = Integral(h(x), x(A), x(B))
f2 = If(x ≥ x(A), If[x ≤ x(B), f(x)])