 Hello and welcome to this screencast on section 9.8, arc length and curvature. This screencast is going to cover arc length and parameterizing with respect to arc length. Suppose we want to know the length of a curve over a given interval. We can use a familiar procedure where we partition the interval into n equals subintervals. On each subinterval, we then approximate the length of the curve by the length of a line segment. Then, to approximate the length of the entire curve, we use the sum of the lengths of all the line segments. For example, here we approximate the length of the curve with one subinterval. Then with two subintervals, with three subintervals, and so on. As the subintervals get smaller and smaller, the approximation gets closer and closer to the actual length of the curve. This procedure gets us to the following formula for arc length. If R defines a smooth curve C on an interval from A to B, then the length of C is given by the integral from A to B of the magnitude of the derivative of R. Note that this formula applies to curves in any dimensional space. Moreover, this formula has a natural interpretation. If R of T records the position of a moving object, then its derivative, R prime, is the object's velocity. And the magnitude of R prime is then its speed. So this formula says that if we simply integrate the speed of an object traveling over the curve to find the distance traveled by the object. In addition to helping us find the length of space curves, the expression for arc length of a curve enables us to find a natural parameterization of space curves in terms of arc length. Consider the parabola y equal to x squared over 2. This curve may be parameterized by the vector-valued function R with x component T and y component T squared over 2. Here, this parameterization is graphed on the left. Notice that as we increment values of T, the points are not equally spaced on the curve. A more natural parameter describing points along the space curve is the distance traveled as we move along the parabola starting at the origin. For instance, on the right graph shows points corresponding to various values of distance traveled denoted by S. We call this an arc length parameterization. To see the benefits of such a parameterization, consider an interstate highway. One way to parameterize the curve defined by the highway is to drive along the highway and record our position at every time, thus creating a function. If we encounter an accident or road construction, this parameterization might not be relevant to another person driving along the same highway. On the other hand, an arc length parameterization is like using the mile markers on the side of the road to specify our position on the highway. If we know how far we've traveled along the highway, then we know exactly where we are.