An arc can be approximated by adding up small line
segments, using the distance formula 2 2
2 1 2 1( ) ( )d x x y y
A rectifiable curve is one that has finite arc length. It is also
continuous, and its graph is a smooth curve.
By letting the points on the curve get closer and closer, you
end up with more and more line segments.

2 2
1
n
i i
i
s x y
2
2 2
2
1
n
i
i i
i i
y
x x
x
Factor out and take square root, and you get
2
ix
2
1
1 ( )
n
i
i
i i
y
x
x
Which results in the definition of arc length: Let y = f(x)
represent a smooth curve on the interval [a, b].
2
1 [ ]
b
a
s f x dx

Similarly, for a smooth curve given by x = g(y), the arc
length of g between c and d is
2
1 [ ]
d
c
s f y dy

Examples: Find the arc length of the graph of
3
1
6 2
x
y
x
in the interval [0.5, 2]

Example: Find the arc length of the graph of
3 2
1y x
from x = 0 to x = 8, (Hint: solve for x, change over to y-
limits