Figure 6.37 depicts this construct for n = 5. 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. Then, for i = 1, 2 ,…, n, i = 1, 2 ,…, n, construct a line segment from the point ( x i − 1, f ( x i − 1 ) ) ( x i − 1, f ( x i − 1 ) ) to the point ( x i, f ( x i ) ). For i = 0, 1, 2 ,…, n, i = 0, 1, 2 ,…, n, let P = be a regular partition of. We start by using line segments to approximate the length of the curve.

We want to calculate the length of the curve from the point ( a, f ( a ) ) ( a, f ( a ) ) to the point ( b, f ( b ) ). Let f ( x ) f ( x ) be a smooth function defined over. (This property comes up again in later chapters.) Functions like this, which have continuous derivatives, are called smooth. Here, we require f ( x ) f ( x ) to be differentiable, and furthermore we require its derivative, f ′ ( x ), f ′ ( x ), to be continuous. However, for calculating arc length we have a more stringent requirement for f ( x ). In previous applications of integration, we required the function f ( x ) f ( x ) to be integrable, or at most continuous. (The process is identical, with the roles of x x and y y reversed.) The techniques we use to find arc length can be extended to find the surface area of a surface of revolution, and we close the section with an examination of this concept. We begin by calculating the arc length of curves defined as functions of x, x, then we examine the same process for curves defined as functions of y. Or, if a curve on a map represents a road, we might want to know how far we have to drive to reach our destination. If a rocket is launched along a parabolic path, we might want to know how far the rocket travels. Many real-world applications involve arc length.

We can think of arc length as the distance you would travel if you were walking along the path of the curve. In this section, we use definite integrals to find the arc length of a curve.

6.4.3 Find the surface area of a solid of revolution.

6.4.2 Determine the length of a curve, x = g ( y ), x = g ( y ), between two points.

6.4.1 Determine the length of a curve, y = f ( x ), y = f ( x ), between two points.

Arc length function