Section 9.3: Arc Length
There are three formulas which give the length of a curve, depending on the variable of
integration.
1. If y = f (x), then the length of the curve from x = a to x = b is given by
Z bp
L=
1 + [f 0 (x)]2 dx.
a
2. If x = g(y), then the length of the curve from y = c to y = d is given by
Z dp
L=
1 + [g 0 (y)]2 dy.
c
3. If x = x(t) and y = y(t), then the length of the curve from t = α to t = β is given by
s
Z β 2 2
dx
dy
L=
+
dt.
dt
dt
α
Example: Find the length of the curve y = 2x3/2 , 0 ≤ x ≤ 1.
√
First, y 0 = 3 x. Then
Z
1
L=
Z
q
√ 2
1 + (3 x) dx =
1
√
1 + 9xdx.
0
0
Let u = 1 + 9x, du = 9dx. Then
Z
1 10 √
L =
udu
9 1
10
1 2 3/2 u
=
9 3
1
2 √
10 10 − 1 .
=
27
Example: Find the length of the curve x = ln(cos y), 0 ≤ y ≤
First, x0 = −
sin y
= − tan y. Then
cos y
Z
π/4
L =
p
1 + tan2 ydy
p
sec2 ydy
0
Z
π/4
=
0
Z
=
π/4
sec ydy
0
= ln | sec y + tan y||π/4
0
√
= ln | 2 + 1|.
π
.
4
Example: Find the length of the curve defined by x = 3t − t3 , y = 3t2 , 0 ≤ t ≤ 2.
First, x0 (t) = 3 − 3t2 and y 0 (t) = 6t. Then
Z 2p
(3 − 3t2 )2 + (6t)2 dt
L =
Z0 2 √
=
9 − 18t2 + 9t4 + 36t2 dt
0
Z 2√
9t4 + 18t2 + 9dt
=
Z0 2 p
=
9(t2 + 1)2 dt
Z0 2
(3t2 + 3)dt
=
0
2
= (t3 + 3t)0
= 14.
Example: Find the length of the curve y =
First, y 0 =
x3
1
+ , 1 ≤ x ≤ 2.
6
2x
1
x2
− 2 . Then
2
2x
0 2
1 + (y )
2
x2
1
1+
− 2
2
2x
4
1
1
x
− + 4
1+
4
2 4x
x4 1
1
+ + 4
4
2 4x
2
2
x
1
+ 2 .
2
2x
=
=
=
=
Thus the length of the curve is
2
x2
1
L =
+ 2 dx
2
2x
1
2
x3
1
=
− 6
2x 1
17
=
.
12
Z
Example: A plane flying at 200 m/s drops a bomb at an altitude of 4500 m. The parabolic
trajectory of the falling bomb is described by
x2
y = 4500 −
8000
until it hits the ground, where y is the altitude and x is the horizontal distance traveled in
meters. Set up an integral that gives the distance traveled by the bomb from the time it is
dropped until the time it hits the ground.
To find the value of x at which the bomb hits the ground, set
y = 0
2
x
= 4500
8000
x2 = 36, 000, 000
x = 6000.
Then the distance traveled is given by
Z
L=
6000
r
1+
0
x 2
dx.
4000