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Integrals and Summations in Physical Systems | Up
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Figure 8 Graph of Velocity vs. Time |
Higher Concepts of Mathematics
CALCULUS
W
x2
x1
F dx
The physical meaning of this equation can be stated in terms of a
summation. The total amount of work done equals the integral of
F dx from x = x1 to x = x2. This can be visualized as taking the
product of the instantaneous force, F, and the incremental change
in position dx at each point between x1 and x2, and summing all
of these products.
2.
Give the physical interpretation of the following equation relating
the amount of radioactive material present as a function of the
elapsed time, t, and the decay constant, l.
N1
N0
dN
N
lt
The physical meaning of this equation can be stated in terms of a
summation. The negative of the product of the decay constant, l,
and the elapsed time, t, equals the integral of dN/N from N = N0
to n = n1. This integral can be visualized as taking the quotient
of the incremental change in N, divided by the value of N at each
point between N0 and N1, and summing all of these quotients.
Graphical Understanding of Integral
As with derivatives, when a functional relationship is presented in graphical form, an important
understanding 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 tA and tB equals the integral of the velocity, v, with
respect to time between the limits tA and tB.
(5-15)
s
tB
tA
v dt
Rev. 0
Page 43
MA-05
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