




To calculate the distance, S, along a curve C between points A and B.
This distance is called arc length of C between A and B.
Let Ds be the distance along the curve between M and N and Dx, Dy their difference in coordinates.
When M and N are very close to each other,
and by the Pythagorean theorem we get
.
We approximate S by a succession of chords, so that the sum of their lengths tends to S as their number increases.
The length S becomes the integral of ds from A to B.
Length S
If the curve C is expressed by parametric equations x(t), y(t):
If the curve C is expressed by y = f(x):
Examples:
Circle
The parametric equations of a circle of radius b are
Calculate the arc length S of the circle.
Astroid
The parametric equations of an astroid are
x = cos^{3}t
y = sin^{3}t
Calculate the arc length of 1 / 4 of the astroid (0 t / 2).
Cycloid
A cycloid is the curve traced out by a point on the circumference of a circle when the circle rolls along a straight line in its own plane. The equations of a cycloid created by a circle of radius 1 are
x(t) = t  sin t
y(t) = 1  cos t
Calculate the arc length S of the cycloid for 0 t / 2.
Ball trajectory
The equations of the trajectory of a thrown ball are
x = at
y = bt  16t^{2}
Example 5
Calculate the arc length of y = x^{2} between x = 0 and x = 2.
Example 6
Calculate the arc length of the curve y = x^{3/2} between x = 0 and x = 1.