CALCULUSHigher Concepts of MathematicsThe value of this integral can be determined forFigure 8 Graph of Velocity vs. Timethe case plotted in Figure 8 by noting that thevelocity is increasing linearly. Thus, the averagevelocity for the time interval between t_{A} and t_{B} isthe arithmetic average of the velocity at t_{A} andthe velocity at t_{B}. At time t_{A}, v= 6t_{A}; at time t_{B},v= 6t_{B}. Thus, the average velocity for the timeinterval between t_{A} and t_{B} is which6t_{A } 6t_{B}2equals 3(t_{A} + t_{B}). Using this average velocity, thetotal distance traveled in the time intervalbetween t_{A} and t_{B} is the product of the elapsedtime t_{B} - t_{A} and the average velocity 3(t_{A} + t_{B}).s= v_{avD}ts= 3(t_{A}+ t_{B})(t_{B}- t_{A})(5-16)Equation 5-16 is also the value of the integral of the velocity, v, with respect to time, t, betweenthe limits t_{A} -t_{B} for the case plotted in Figure 8.t_{B}t_{A}vdt 3(t_{A } t_{B})(t_{B } t_{A})The cross-hatched area in Figure 8 is the area under the velocity curve between t = t_{A} and t =t_{B}. The value of this area can be computed by adding the area of the rectangle whose sides aret_{B}- t_{A}and the velocity at t_{A}, which equals 6t_{A}- t_{B}, and the area of the triangle whose base is t_{B}-t_{A}and whose height is the difference between the velocity at t_{B}and the velocity at t_{A}, whichequals 6t_{B} - t_{A}.Area [(t_{B } t_{A})(6t_{A})] 12(t_{B } t_{A})(6t_{b } 6t_{A})Area 6t_{A} t_{B } 6t^{2}_{A } 3t^{2}_{B } 6t_{A} t_{B } 3t^{2}_{A}Area 3t^{2}_{B } 3t^{2}_{A}Area 3(t_{B } t_{A})(t_{B } t_{A})MA-05 Page 44Rev. 0

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