Arc Length

Arc Length in Rectangular Coordinates

Let a curve C be defined by the equation y = f (x) where f is continuous on an interval [a, b]. We will assume that the derivative f '(x) is also continuous on [a, b].

The length of the curve $y=f\left(x\right)$ from $x=a$ to $x=b$ is given by

If we use Leibniz notation for derivatives, the arc length is expressed by the formula

We can introduce a function that measures the arc length of a curve from a fixed point of the curve. Let $P_0\left(a,f\left(a\right)\right)$ be the initial point on the curve $y=f\left(x\right),$ $a\le x\le b.$ Then the arc length of the curve from $P_0\left(a,f\left(a\right)\right)$ to a point $Q\left(x,f\left(x\right)\right)$ is given by the integral

where $t$ is an internal variable of the integral.

The function $S\left(x\right)$ is called the arc length function.

Arc Length of a Parametric Curve

If a curve $C$ is given in parametric form by the equations

where the parameter $t$ runs between $t_1$ and $t_2,$ the arc length of the curve is

Arc Length in Polar Coordinates

The arc length of a polar curve given by the equation $$ with $$ ranging over some interval $$ is expressed by the formula

Solved Problems

Solution.

Let's solve the more general problem. Consider an arbitrary straight line that is defined by the equation $$ What is the length of the line segment in the interval $$

It's obvious that

Using the arc length formula, we have

So, for the line segment $$ we obtain

Solution.

We use the arc length formula

Here $$ $$ $$ $$ Then

We make the substitution

When $$ $$ and when $$ $$ So the integral becomes

Solution.

We calculate the circumference of the upper half of the circle and then multiply the answer by $$ The upper half of the circle is defined by the function

Take the derivative:

Then the circumference of the circle is given by

Solution.

Since $$ we can write

To evaluate the latter integral we rewrite it in the form

and change the variable:

This results in

Returning back to the definite integral, we find the arc length $$