Higher Concepts of MathematicsCALCULUSW x_{2}x_{1}F dxThe physical meaning of this equation can be stated in terms of asummation. The total amount of work done equals the integral ofFdxfrom x= x_{1}to x= x_{2}. This can be visualized as taking theproduct of the instantaneous force, F, and the incremental changein position dx at each point between x_{1} and x_{2}, and summing allof these products.2.Give the physical interpretation of the following equation relatingthe amount of radioactive material present as a function of theelapsed time, t, and the decay constant, l.N_{1}N_{0}dNNltThe physical meaning of this equation can be stated in terms of asummation. The negative of the product of the decay constant, l,and the elapsed time, t, equals the integral of dN/N from N = N_{0}to n = n_{1}. This integral can be visualized as taking the quotientof the incremental change in N, divided by the value of N at eachpoint between N_{0} and N_{1}, and summing all of these quotients.GraphicalUnderstandingofIntegralAs with derivatives, when a functional relationship is presented in graphical form, an importantunderstanding of the meaning of integral can be developed.Figure 8 is a plot of the instantaneous velocity, v, of an object as a function of elapsed time, t.The functional relationship shown is given by the following equation:v= 6t(5-14)The distance traveled, s, between times t_{A} and t_{B}equals the integral of the velocity, v, withrespect to time between the limits t_{A} and t_{B}.(5-15)s t_{B}t_{A}v dtRev. 0 Page 43MA-05