Arc Length of the Curve y = f ( x )

In previous applications of integration, we required the function \(f \left(\right. x \left.\right)\) to be integrable, or at most continuous. However, for calculating arc length we have a more stringent requirement for \(f \left(\right. x \left.\right) .\) Here, we require \(f \left(\right. x \left.\right)\) to be differentiable, and furthermore we require its derivative, \(f^{'} \left(\right. x \left.\right) ,\) to be continuous. Functions like this, which have continuous derivatives, are called smooth. (This property comes up again in later chapters.)

Let \(f \left(\right. x \left.\right)\) be a smooth function defined over \(\left[\right. a , b \left]\right. .\) We want to calculate the length of the curve from the point \(\left(\right. a , f \left(\right. a \left.\right) \left.\right)\) to the point \(\left(\right. b , f \left(\right. b \left.\right) \left.\right) .\) We start by using line segments to approximate the length of the curve. For \(i = 0 , 1 , 2 ,\ldots , n ,\) let \(P = \left{\right. x_{i} \left.\right}\) be a regular partition of \(\left[\right. a , b \left]\right. .\) Then, for \(i = 1 , 2 ,\ldots , n ,\) construct a line segment from the point \(\left(\right. x_{i - 1} , f \left(\right. x_{i - 1} \left.\right) \left.\right)\) to the point \(\left(\right. x_{i} , f \left(\right. x_{i} \left.\right) \left.\right) .\) Although it might seem logical to use either horizontal or vertical line segments, we want our line segments to approximate the curve as closely as possible. Figure 6.37 depicts this construct for \(n = 5 .\)

Figure 6.37 We can approximate the length of a curve by adding line segments.

To help us find the length of each line segment, we look at the change in vertical distance as well as the change in horizontal distance over each interval. Because we have used a regular partition, the change in horizontal distance over each interval is given by \(\Delta x .\) The change in vertical distance varies from interval to interval, though, so we use \(\Delta y_{i} = f \left(\right. x_{i} \left.\right) - f \left(\right. x_{i - 1} \left.\right)\) to represent the change in vertical distance over the interval \(\left[\right. x_{i - 1} , x_{i} \left]\right. ,\) as shown in Figure 6.38. Note that some (or all) \(\Delta y_{i}\) may be negative.

Figure 6.38 A representative line segment approximates the curve over the interval \(\left[\right. x_{i - 1} , x_{i} \left]\right. .\)

By the Pythagorean theorem, the length of the line segment is \(\sqrt{\left(\right. \Delta x \left.\right)^{2} + \left(\right. \Delta y_{i} \left.\right)^{2}} .\) We can also write this as \(\Delta x \sqrt{1 + \left(\right. \left(\right. \Delta y_{i} \left.\right) / \left(\right. \Delta x \left.\right) \left.\right)^{2}} .\) Now, by the Mean Value Theorem, there is a point \(x_{i}^{\star} \in \left[\right. x_{i - 1} , x_{i} \left]\right.\) such that \(f^{'} \left(\right. x_{i}^{\star} \left.\right) = \left(\right. \Delta y_{i} \left.\right) / \left(\right. \Delta x \left.\right) .\) Then the length of the line segment is given by \(\Delta x \sqrt{1 + \left[\right. f^{'} \left(\right. x_{i}^{\star} \left.\right) \left]\right.^{2}} .\) Adding up the lengths of all the line segments, we get

This is a Riemann sum. Taking the limit as \(n \rightarrow \infty ,\) we have

We summarize these findings in the following theorem.

Note that we are integrating an expression involving \(f^{'} \left(\right. x \left.\right) ,\) so we need to be sure \(f^{'} \left(\right. x \left.\right)\) is integrable. This is why we require \(f \left(\right. x \left.\right)\) to be smooth. The following example shows how to apply the theorem.

Although it is nice to have a formula for calculating arc length, this particular theorem can generate expressions that are difficult to integrate. We study some techniques for integration in Introduction to Techniques of Integration. In some cases, we may have to use a computer or calculator to approximate the value of the integral.