Introduction
We calculate the distance between two points on a function
(a, f(a))
,
(b, f(b))
, as the bird flies:
\sqrt{(b - a)^2 + (f(b) - f(a))^2}
We can also calculate the distance between two points on a function while travelling along the function.
This is the arc length.
In other words, the euclidean distance is the green line. The arc length is the red line.
Calculation
The formula above is the same as the one below: Euclidean Distance:
D = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\Delta x
is the change in
x
\Delta y
is the change in
y
.
We calculate the arc length between two points by summing the euclidean distances of all the points along the function between them:
L =\sum_{i=1}^{n} \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2}
This Riemann Sum can be converted to an integral:
\begin{align*}
L &= \int_{a}^{b} \sqrt{dx^2 + dy^2} \\
&= \int_{a}^{b} \sqrt{dx^2 + dy^2} \cdot \frac{dx}{dx} \\
&= \int_{a}^{b} \sqrt{\frac{dx^2}{dx^2} + \frac{dy^2}{dx^2}} \, dx \\
&= \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\end{align*}
Examples
Example 1: Explicit Function
Find the arc length of
y = \frac{2}{3}x^{3/2}
from
x = 0
to
x = 3
.
\begin{align*}
\frac{dy}{dx} &= \sqrt{x} \\
L &= \int_{0}^{3} \sqrt{1 + (\sqrt{x})^2} \, dx \\
&= \int_{0}^{3} \sqrt{1 + x} \, \, dx \\
&= \left[ \frac{2}{3}(1 + x)^{3/2} \right]_0^3 \\
&= \frac{2}{3}(8 - 1) = \frac{14}{3} \approx 4.67 \, \text{units}
\end{align*}
The function travels a distance of
\frac{14}{3}
units.
Example 2: Linear Function Verification
Find the arc length of
y = 3x + 2
from
x = 1
to
x = 4
.
\begin{align*}
\frac{dy}{dx} &= 3 \\
L &= \int_{1}^{4} \sqrt{1 + 3^2} \, dx \\
&= \int_{1}^{4} \sqrt{10} \, dx \\
&= \sqrt{10}(4 - 1) = 3\sqrt{10} \approx 9.49 \, \text{units}
\end{align*}
Since this function is a straight line, we can check our work using the euclidean distance between
(1, 5)
and
(4, 14)
:
\sqrt{(4-1)^2 + (14-5)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}
Example 3: Simplifying Integral
Find the arc length of
y = \frac{1}{3}(x^2 + 2)^{3 / 2}
from
x = 0
to
x = 1
.
\begin{align*}
\frac{dy}{dx} &= x(x^2 + 2)^{1/2} \\
L &= \int_{0}^{1} \sqrt{1 + \left(x\sqrt{x^2 + 2}\right)^2} \, dx \\
&= \int_{0}^{1} \sqrt{1 + x^2(x^2 + 2)} \, dx \\
&= \int_{0}^{1} \sqrt{x^4 + 2x^2 + 1} \, dx \\
&= \int_{0}^{1} (x^2 + 1) \, dx \\
&= \left[\frac{x^3}{3} + x\right]_0^1 = \frac{4}{3} \, \text{units}
\end{align*}
The arc length is
\frac{4}{3}
units.