Applications of Integration: Arc Length. Dr. Dillon Calculus II Fall 1999 SPSU. Start with something easy. The length of the line segment joining points (x 0 ,y 0 ) and (x 1 ,y 1 ) is. (x 1 ,y 1 ). (x 0 ,y 0 ). The Length of a Polygonal Path?. Add the lengths of the line segments.
The length of the line segment joining points (x0,y0) and (x1,y1) is
Add the lengths of the line segments.
Approximate by chopping it into polygonal pieces and adding up the lengths of the pieces
which is the approximate length of the curve
We can always view y as a function of x, at least locally (just looking at one little piece of the curve)
And if you don’t buy that…
we can view x as a function of y when we can’t view y as a function of x...
Assume that y is a function of x
and that y is differentiable
with a continuous derivative
The length of the curve is approximately
inside the radical to get
And from there
That gives us
Where the limit is taken over all partitions
Playing the role of F(xi) we have
And to make things more interesting
The x coordinates of the endpoints of the arc
One way to deal with that would be to treat the arc in sections.
Find the length of the each section, then add.
If a curve is described by y=f(x) on the interval [a,b]
then the length L of the curve is given by