MTH 122 — Calculus II
Essex County College — Division of Mathematics and Physics
1
Lecture Notes #10 — Sakai Web Project Material
1
Arc Length
Everyone should be familiar with the distance formula that was introduced in elementary algebra.
It is a basic formula for the linear distance between two points in the plane. It states that the
distance between (
x
1
, y
1
) and (
x
2
, y
2
) is
d
=
q
(
x
2

x
1
)
2
+ (
y
2

y
1
)
2
.
This distance, of course, is for a line connecting those two points. However, what if we have a
curve and we want to know the distance along that curve between two points? We will basically
cut the curve into an infinite number of small linear sections, and then add these sections together
to get the arc length, or distance between two points on the curve. Here a definite integral can
be used to find the arc length, where we have a curve,
f
(
x
), and two points on this curve that
are connected by a curve that is continuously differentiable on the interval.
Arc Length Formula:
If
f
0
is continuous on [
a, b
], then the length of the curve
y
=
f
(
x
),
a
≤
x
≤
b
, is
L
=
Z
b
a
q
1 + [
f
0
(
x
)]
2
d
x.
This can also be written as
L
=
Z
b
a
s
1 +
d
y
d
x
2
d
x.