Graphical Understanding of Integral:

As with derivatives, while a functional relationship is presented in graphical form, an important understanding of the meaning of integral could be developed.

Figure is a plot of the instantaneous velocity, v, of an object since a function of elapsed time, t. The functional relationship display is given by the following equation:

v = 6t

The distance traveled, s, among times t_A   and t_B equals the integral of the velocity, v, with respect to time between the limits t_A and t_B.

The value of this integral can be determined for the case plotted in Figure through noting that the velocity is increasing linearly.  Therefore, the average velocity for the time interval among t_A  and t_B  is the arithmetic  average  of the velocity  at t_A   and the velocity at t_B.  At time t_A, v = 6tA; at time t_B, v = 6t_B.  Therefore, the average velocity for the time interval among t_A and t_B is 6tA +6t_B/2 that equals 3(t_A  + t_B).  Using this average velocity, the total   distance   traveled   in   the   time   interval between t_A   and t_B is the product of the elapsed time t_B - t_A and the average velocity 3(t_A  + t_B).

s = v _avΔt

s = 3(t_A+ t_B)(t_B - t_A)

Figure: Graph of Velocity vs. Time