Because a parabolic curve becomes sharp gradually, as shown in the first figure on the nex...

Because a parabolic curve becomes sharp gradually, as shown in the first figure on the next page, curves designed by engineers for highways and railroads frequently have parabolic, rather than circular, shapes. If railroad tracks changed abruptly from straight to circular, the momentum of the locomotive could cause a derailment. The second figure on the next page illustrates straight tracks connecting to a circular curve. (Source: F. Mannering and W. Kilareski, Principles of Highway Engineering and Traffic Analysis.)

In order to design a curve and estimate its cost, engineers determine the distance around the curve before it is built. In the third figure on the next page the distance along a parabolic curve from A to C is approximated by two line segments AB and BC. The distance formula can be used to calculate the length of each segment. The sum of these two lengths gives a crude estimate of the length of the curve.

A better estimate can be made using four line segments, as shown in the fourth figure. As the number of segments increases, so does the accuracy of the approximation.

1. Curve Length Suppose that a curve designed for rail-road tracks is represented by the equation y = 0.2x2, where the units are in kilometers. The points (−3, 1.8), (−1.5, 0.45), (0, 0), (1.5, 0.45), and (3, 1.8) lie on the graph of y = 0.2x2. Approximate the length of the curve from x = −3 to x = 3 by using line segments connecting these points.