We next recall a general principle that will later be applied to distance-velocity-acceleration problems, among other things. If is an anti-derivative of , then . Suppose that we want to let the upper limit of integration vary, i.e., we replace by some variable . We think of as a fixed starting value . In this new notation the last equation (after adding to both sides) becomes: (Here is the variable of integration, called a "dummy variable,'' since it is not the variable in the function . In general, it is not a good idea to use the same letter as a variable of integration and as a limit of integration. That is, is bad notation, and can lead to errors and confusion.)
An important application of this principle occurs when we are interested in the position of an object at time (say, on the -axis) and we know its position at time . Let denote the position of the object at time (its distance from a reference point, such as the origin on the -axis). Then the net change in position between and is . Since is an anti-derivative of the velocity function , we can write Similarly, since the velocity is an anti-derivative of the acceleration function , we have
Suppose an object is acted upon by a constant force . Find and . By Newton's law , so the acceleration is , where is the mass of the object. Then we first have using the usual convention . Then For instance, when is the constant of gravitational acceleration, then this is the falling body formula (if we neglect air resistance) familiar from elementary physics: or in the common case that ,
Recall that the integral of the velocity function gives the net distance traveled. If you want to know the total distance traveled, you must find out where the velocity function crosses the -axis, integrate separately over the time intervals when is positive and when is negative, and add up the absolute values of the different integrals. For example, if an object is thrown straight upward at 19.6 m/sec, its velocity function is , using m/sec for the force of gravity. This is a straight line which is positive for and negative for . The net distance traveled in the first 4 seconds is thus while the total distance traveled in the first 4 seconds is , meters up and meters down.