Higher Concepts of MathematicsCALCULUSThis is exactly equal to the value of the integral of the velocity with respect to time betweenthe limits t_{A} and t_{B}. Since the distance traveled equals the integral of the velocity with respectto time, vdt, and since this integral equals the area under the curve of velocity versus time, thedistance 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 ofa 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 andthe graphical plot of the function.f(x)dx= Area between f(x) and x-axis from x_{1}to x_{2}X_{2}X_{1}The mathematics of dynamic systems involves many different operations with the integral offunctions. As with derivatives, in practice, the integral of functions are not determined byplotting the functions and measuring the area under the curves. Although this approach couldbe used, techniques have been developed which permit integral of functions to be determineddirectly based on the form of the functions. Actually, the technique for taking an integral is thereverse of taking a derivative. For example, the derivative of the function f(x) = ax + c, whereaand care constants, is a. The integral of the function f(x) = a, where ais a constant, is ax+c, where aand care constants.f(x) = a(5-17)f(x)dx ax cThe integral of the function f(x) = ax^{n}, where a and n are constants, is , wherean 1xn1ccis another constant.f(x) = ax^{n}(5-18)f(x)dx an 1xn 1cThe integral of the function f(x) = ae^{bx}, where a and b are constants and e is the base of naturallogarithms, is , where c is another constant.ae^{bx}bcRev. 0 Page 45MA-05