Higher Concepts of MathematicsCALCULUSIntegralsand Summationsin PhysicalSystemsDifferentials and derivatives arose in physical systems when small changes in one quantity wereconsidered. For example, the relationship between position and time for a moving object led tothe definition of the instantaneous velocity, as the derivative of the distance traveled with respectto time, ds/dt. In many physical systems, rates of change are measured directly. Solvingproblems, when this is the case, involves another aspect of the mathematics of dynamic systems;namely integral and summations.Figure 7 is a graph of the instantaneous velocity of an object as a function of elapsed time. Thisis the type of graph which could be generated if the reading of the speedometer of a car wererecorded as a function of time.At any given instant of time, the velocityFigure 7 Graph of Velocity vs. Timeof the object can be determined byreferring to Figure 7. However, if thedistance traveled in a certain interval oftime is to be determined, some newtechniques must be used. Consider thevelocity versus time curve of Figure 7.Let's consider the velocity changesbetween times t_{A} and t_{B}. The firstapproach is to divide the time interval intothree short intervals , and to(D t_{1},D t_{2},D t_{3})assume that the velocity is constant duringeach of these intervals. During timeinterval Dt_{1}, the velocity is assumedconstant at an average velocity v_{1}; duringthe interval Dt_{2}, the velocity is assumedconstant at an average velocityv_{2}; duringtime interval Dt_{3}, the velocity is assumedconstant at an average velocityv_{3}. Thenthe total distance traveled is approximately the sum of the products of the velocity and theelapsed time over each of the three intervals. Equation 5-10 approximates the distance traveledduring the time interval from t_{a}to t_{b} and represents the approximate area under the velocity curveduring this same time interval.s= v_{1D}t_{1}+ v_{2D}t_{2}+ v_{3D}t_{3}(5-10)Rev. 0 Page 41MA-05