Notes 11: Arc Length
If we have a parameterization
(
x = f (t )
y = g (t )
then we can interpret ẋ as the horizontal component of the velocity and ẏ as the vertical component of the
velocity. By Pythagorus’
Theorem, the speed of a particle moving along the curve according to the paramep
terization is (ẋ)2 + ( ẏ)2 . One of the main uses for this formula is to find the total distance that the particle
traveled along the curve from one point to another. This is called the arc length (or just length) between the
starting and ending point. It is often denoted by s and is given by
Z bq
s=
(ẋ)2 + ( ẏ)2 d t ,
a
where t = a and t = b are the two values that correspond to the starting and ending points.
Example 1: Find the arc length of the curve parameterized by
(
x = 2t 3/2
y =t
for 1 ≤ t ≤ 83 .
Example 2: Find the arc length of the curve parameterized by
(
x = e t cos(t )
y = e t sin(t )
for 0 ≤ t ≤ 2π.
Example 3: Find the arc length of the curve parameterized by
(
x = 21 t 2 − ln(t )
y = 2t
for 1 ≤ t ≤ 2.
Example 4: Find the arc length of the cycloid parameterized by
(
x = r (t − sin(t ))
y = r (1 − cos(t ))
for 0 ≤ t ≤ 2π.
Example 5: Find the arc length of the Euler spiral parameterized by
(
Rt
x = 0 cos(θ 2 ) d θ
Rt
y = 0 sin(θ 2 ) d θ
for 0 ≤ t ≤ 10.
Example 6: Find the arc length of the arc of the parabola 2y = x 2 from (0, 0) to (2, 2).