Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

1. Applications of Integration:
Arc Length
Dr. Dillon
Calculus II
Fall 1999
SPSU

2. Start with something easy
The length of the line segment joining
points (x0,y0) and (x1,y1) is
2
10
2
10 )()( yyxx −+−
(x1,y1)
(x0,y0)

3. The Length of a Polygonal
Path?
Add the lengths of the line segments.

4. The length of a curve?
Approximate by chopping it into polygonal pieces
and adding up the lengths of the pieces

5. Approximate the curve with
polygonal pieces?

6. What is the approximate
length of your curve?
• Say there are n line segments
– our example has 18
• The ith
segment connects (xi-1, yi-1) and (xi,
yi)
(xi-1,yi-1) (xi, yi)

7. The length of that ith
segment
is...
2
1
2
1 )()( iiii yyxx −+− −−

8. The length of the polygonal
path is thus...
which is the approximate length of the
curve
∑=
−− −+−
n
i
iiii yyxx
1
2
1
2
1 )()(

9. What do we do to get the
actual length of the curve?
• The idea is to get the length of the curve
in terms of an equation which describes
the curve.
• Note that our approximation improves
when we take more polygonal pieces

10. For Ease of Calculation...
iii xxx −=∆ −1
iii yyy −=∆ −1
Let
and

11. A Basic Assumption...
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...

12. To keep our discussion
simple...
Assume that y is a function of x
and that y is differentiable
with a continuous derivative

13. Using the delta notation, we
now have…
The length of the curve is approximately
∑=
∆+∆
n
i
ii yx
1
22
)()(

14. Simplify the summands...
Factor out
))/(1()( 22
iii xyx ∆∆+∆
And from thereAnd from there
2
)/(1 iii xyx ∆∆+∆
2
)( ix∆ inside the radical to get

15. Now the approximate arc
length looks like...
i
n
i
ii xxy ∆∆∆+∑=1
2
)/(1

16. To get the actual arc length L?
Let 0→∆ ix
That gives usThat gives us
dxdxdyL
b
a
∫ += 2
)/(1

17. What? Where’d you get that?
)()(
10
lim ∫∑ =∆
=→∆
b
a
n
i
ii
x
dxxFxxF
i
Where the limit is taken over all partitionsWhere the limit is taken over all partitions
{ } [ ]baxx n ,intervaltheof,...,0
AndAnd 1−−=∆ iii xxx
Recall that

18. In this setting...
Playing the role of F(xi) we have
( )2
)/(1 ii xy ∆∆+
And to make things more interestingAnd to make things more interesting
than usual,than usual,
dxdyxy ii
xi
//lim0
=∆∆
→∆

19. What are a and b?
The x coordinates of the endpoints of the arc

20. Endpoints? Our arc crossed
over itself!
One way to deal with that would be to treat the arc
in sections.
Find the length of the each section, then add.
a b

21. Conclusion?
If a curve is described by y=f(x) on the interval
[a,b]
then the length L of the curve is given by
( ) dxxfL
b
a
))('(1 2
∫ +=