Higher Concepts of Mathematics
Integrals and Summations in Physical Systems
Differentials and derivatives arose in physical systems when small changes in one quantity were
considered. For example, the relationship between position and time for a moving object led to
the definition of the instantaneous velocity, as the derivative of the distance traveled with respect
to time, ds/dt. In many physical systems, rates of change are measured directly. Solving
problems, 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. This
is the type of graph which could be generated if the reading of the speedometer of a car were
recorded as a function of time.
At any given instant of time, the velocity
Figure 7 Graph of Velocity vs. Time
of the object can be determined by
referring to Figure 7. However, if the
distance traveled in a certain interval of
time is to be determined, some new
techniques must be used. Consider the
velocity versus time curve of Figure 7.
Let's consider the velocity changes
between times tA and tB.
approach is to divide the time interval into
three short intervals
, and to
(D t1,D t2,D t3)
assume that the velocity is constant during
each of these intervals.
interval Dt1, the velocity is assumed
constant at an average velocity v1; during
the interval Dt2, the velocity is assumed
constant at an average velocity v2; during
time interval Dt3, the velocity is assumed
constant at an average velocity v3. Then
the total distance traveled is approximately the sum of the products of the velocity and the
elapsed time over each of the three intervals. Equation 5-10 approximates the distance traveled
during the time interval from ta to tb and represents the approximate area under the velocity curve
during this same time interval.
s = v1Dt1 + v2Dt2 + v3Dt3