In this video I go over a pretty in depth proof of the arc length or length of a curve as well as deriving a formula for it as an integral. The length of a curve can be defined in a precise way in much the same way as with other applications of integrals, such as Area or Volume. The process of deriving its formula is simply approximating it as a sum of polygons and then taking a limit as the number of polygons goes to infinite. The fact that integrals can be used to precisely determine the length of any continuous curve goes to show how powerful integral calculus can be!

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Applications of Integrals: Arc Length

What do we mean by the length of a curve?

We might think of fitting a piece of string to the curve in the figure below and measuring the string against a ruler.

But this method is too difficult to do with much accuracy if we have a complicated curve.

Instead we need a precise definition for the length of an arc of a curve, in the same way we developed for the concepts of area and volume (as shown in my earlier videos).

If the curve is a polygon, we can easily find its length; we just add the lengths of the line segments that form the polygon.

Thus to define the length of a general curve, we will first approximate it by a polygon and then taking a limit as the number of segments of the polygon increase.

This process is the same for the case of a circle, where the circumference is the limit of lengths of inscribed polygons, as shown below. (Also see my earlier video on the proof of π which I use this same method to prove.)

Now suppose that a curve C is defined by the equation y = f(x), where f is continuous on the interval a ≤ x ≤ b.

We can obtain a polygonal approximation to C by dividing the interval [a, b] into n subintervals of equal width Δx and endpoints x₀, x₁, … , x_n.

If y_i = f(x_i), then the point P_i (x_i , y_i) lies on C and the polygon with vertices P₀, P₁, … , P_n is an approximation to C.

The length L of C is approximately the length of this polygon and the approximation gets betters as we let n increase.

We can illustrate this by zooming in on the arc of the curve between P_i-1 and P_i show how the approximation becomes better with smaller values of Δx.

Thus we can define the length L of the curve C as the limit of lengths of these inscribed polygons (if the limit exists).

Where the notation |P_i-1P_i| simply means the distance from point P_i-1 to P_i.

Notice how the procedure for defining arc length is very similar to the procedure we used for defining area and volume:

1. We divided the curve into a large number of small parts.
2. We then found the approximate lengths of the small parts and added them.
3. Finally, we took the limit as n → ∞.

The definition of arc length shown is not very convenient for computational purposes, but we can derive an integral formula for L in the case where f has a continuous derivative.

Such a function f is called smooth because a small change in x produces a small change in f'(x).

Recall the Mean Value Theorem:

1. If f is continuous on the closed interval [a, b]
2. And if f is differentiable on the open interval (a, b)

Then there exists a number c such that the derivative is the average slope of f between a and b.

Recall that this expression is the same expression for the definition of a definite integral.

This integral exists because f and f' are both defined as being continuous, thus the integrand is also continuous.

Thus in summary, we have proved the following theorem:

The Arc Length Formula:

If f and f' are both continuous on [a, b], then the length of the curve y = f(x) on the interval a ≤ x ≤ b is: