Finding arc length by integration

By Martin McBride, 2024-06-15
Tags: line integral arc length
Categories: integration calculus


We often use the technique of integration to find the area under a curve. It is also possible to use integration to find the arc length of a curve. We will explore that in this article.

What is the arc length of a curve?

We know from elementary geometry how to calculate the length of an arc of a circle:

Arc of a circle

An arc is a part of the circumference of the circle. The red line is an example of an arc.

Intuitively, we might measure the length of the arc by placing a length of string along the red part of the circumference of the circle. We could then stretch out the string on a straight line to measure it. Alternatively, of course, we could use the standard formula, where r is the radius of the circle, and θ is the angle that the arc subtends at the centre, measured in radians:

Arc of a circle

The arc length of a curve is quite similar, except that it might be based on a different function. Here is an arc on a quadratic curve:

Arc of quadratic

Again we could measure the approximate length of the arc with string, but for an exact result, we can use integration, as we will see below.

Area integration

To understand how we might use integration to calculate an arc length, it is worth taking a step back and revisiting how we use integration to calculate the area under a curve. Let's look at the shaded area under the curve created by some function f(x):

Area under curve

We can approximate this area by dividing the area into slices, each of width Δx and height f(x). The graphs show the area divided into two slices or four slices:

Area under curve

The area is approximately:

Area under curve

In the graph above, the individual blue rectangles represent the terms in the sum, so the total area of all the rectangles is the approximate area under the curve. The orange, triangular regions represent the error in the approximation. The important thing to notice is that, as the number of slices increases, the total orange area gets smaller, so the area of the blue rectangles gets closer and closer to the exact area under the curve.

As the number of slices increases, and Δx tends to zero, of course, the sum becomes a definite integral that gives the exact area under the curve:

Area under curve

Length of arc - a naive (and incorrect) first attempt

Now let's try to find the arc length of an interval of this curve:

Length of arc

We will try the same technique as before. We will divide the curve into several sub-intervals, each of length Δx, and find the approximate arc length of each sub-interval. Then, hopefully, as we increase the number of slices, the approximation will get more and more accurate.

Let's make the simplest possible assumption, that the length of the arc is approximately equal to the width of the sub-interval, Δx. This is shown here, for two sub-intervals and four sub-intervals:

Length of arc

As Δx gets smaller and smaller, each sub-interval of the arc gets smaller too. But does this mean that the arc lengths will get closer to Δx? Well, unfortunately, no it doesn't.

It is easy to see why. In the first case, with two sub-intervals, the sum of the two Δx values is simply equal to the length of the interval, b - a. When we increase the number of sub-intervals to four, the four intervals, the sum of the lengths is still equal to b - a. No matter how small we make Δx, the total length will remain the same.

Length of arc - correct method

Rather than measuring the horizontal distance, Δx, between the different sub-intervals, we will try measuring the diagonal distance, Δs, between the two points on the curve at the start and end of each sub-interval:

Length of arc

In this case, when we move from two sub-intervals to four sub-intervals, the diagonal lines become more similar to the curve. If we repeat this process with smaller and smaller values of Δx the approximation will converge on the original curve.

How do we find the value of Δs? Well, for each of the diagonal lines we can form a right-angled triangle involving Δx (the change in x since the previous point) and Δy (the change in y since the previous point):

Length of arc

We can use Pythagoras' Theorem to solve this triangle:

Length of arc

This gives Δs as:

Length of arc

To convert this to an integral, it will be useful to get it is to the form of some function of x multiplied by Δx. We do this by dividing the terms under the square root by Δx squared and multiplying the entire expression by Δx. The first term under the square root then cancels to 1:

Length of arc

The total length of the arc can be found by summing the lengths of all the diagonal lines:

Length of arc

As with the area, if we allow Δx to become infinitesimally small, the sum becomes an integral. In this case, the term Δy over Δx becomes the derivative of y (ie the derivative of f(x)):

Length of arc

It really shouldn't surprise us that the derivative of f(x) should appear in the solution. After all, the length of each diagonal line depends on the slope of the line, which is equal to the slope of the curve at that point.

Example - arc length of cosh function

The solution is valid but the integrals it produces aren't particularly nice. We need to integrate the square root of an expression that includes the square of the derivative of our function. It can be messy, but it is what it is.

Most examples of arc length you will find tend to use slightly unusual functions that simplify the square root expression. We will proudly uphold that tradition with our first example, and find the arc length of the hyperbolic cosine function, cosh.

We will start with a standard result - the derivative of cosh is sinh:

Length of arc of cosh

If we substitute this into the previous equation for the arc length as an integral we get:

Length of arc of cosh

We can use the following identity to simplify the integral. The third form is most useful to us:

Length of arc of cosh

This, of course, is why the cosh function is such a nice example. The pesky square root disappears:

Length of arc of cosh

This is another standard result - the integral of cosh is also sinh:

Length of arc of cosh

So we have our result.

Another example

Here is another example. Again, it is a carefully chosen function:

Length of arc of power

Using the standard rule for differentiating a power we have:

Length of arc of power

The original factor of 2/3 was, of course, designed to cancel out with the 3/2 obtained from the differentiation. Substituting this into the arc length formula gives:

Length of arc of power

Again this is designed so that the ones cancel, leaving a simple integral. It would be possible to solve the integral that wasn't designed this way, but it would be more complex and take away from the topic we are trying to explore. Here is the result:

Length of arc of power

Swapping the axes

There is a useful observation that can sometimes be used to simplify the calculations. Consider the arc of f(x) over the interval a to b:

Swapping the axes

The same curve can also be described as a function g(y) where g is the inverse function of f:

Swapping the axes

Since this is the same curve, it will have the same arc length. So instead of calculating the arc length integral for f(x) over the interval a to b, we can alternatively calculate the arc length integral for g(y) over the interval c to d. Here, c is f(a), and d is f(b).

If f(x) is difficult to integrate, it is worth checking if the inverse function is easier to deal with.

See also



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