Higher Concepts of Mathematics
This is exactly equal to the value of the integral of the velocity with respect to time between
the limits tA and tB. Since the distance traveled equals the integral of the velocity with respect
to time, vdt, and since this integral equals the area under the curve of velocity versus time, the
distance traveled can be visualized as the area under the curve of velocity versus time.
For the case shown in Figure 8, the velocity is increasing at a constant rate. When the plot of
a function is not a straight line, the area under the curve is more difficult to determine.
However, it can be shown that the integral of a function equals the area between the x-axis and
the graphical plot of the function.
f(x)dx = Area between f(x) and x-axis from x1 to x2
The mathematics of dynamic systems involves many different operations with the integral of
functions. As with derivatives, in practice, the integral of functions are not determined by
plotting the functions and measuring the area under the curves. Although this approach could
be used, techniques have been developed which permit integral of functions to be determined
directly based on the form of the functions. Actually, the technique for taking an integral is the
reverse of taking a derivative. For example, the derivative of the function f(x) = ax + c, where
a and c are constants, is a. The integral of the function f(x) = a, where a is a constant, is ax +
c, where a and c are constants.
f(x) = a
f(x)dx ax c
The integral of the function f(x) = axn, where a and n are constants, is
c is another constant.
f(x) = axn
The integral of the function f(x) = aebx, where a and b are constants and e is the base of natural
, where c is another constant.