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Integrals and Summations in Physical Systems | Up
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Graphical Understanding of Integral |
CALCULUS
Higher Concepts of Mathematics
This type of expression is called a summation. A summation indicates the sum of a series of
similar quantities. The upper case Greek letter Sigma, , is used to indicate a summation.
Generalized subscripts are used to simplify writing summations. For example, the summation
given in Equation 5-10 would be written in the following manner:
(5-11)
S
3
i 1
viD ti
The number below the summation sign indicates the value of i in the first term of the
summation; the number above the summation sign indicates the value of i in the last term of the
summation.
The summation that results from dividing the time interval into three smaller intervals, as shown
in Figure 7, only approximates the distance traveled. However, if the time interval is divided
into incremental intervals, an exact answer can be obtained. When this is done, the distance
traveled would be written as a summation with an indefinite number of terms.
(5-12)
S
¥
i 1
viD ti
This expression defines an integral. The symbol for an integral is an elongated "s" . Using
an integral, Equation 5-12 would be written in the following manner:
(5-13)
S
tB
tA
v dt
This expression is read as S equals the integral of v dt from t = tA to t = tB. The numbers below
and above the integral sign are the limits of the integral. The limits of an integral indicate the
values at which the summation process, indicated by the integral, begins and ends.
As with differentials and derivatives, one of the most important parts of understanding integrals
is having a physical interpretation of their meaning. For example, when a relationship is written
as an integral, the physical meaning, in terms of a summation, should be readily understood.
In the previous example, the distance traveled between tA and tB was approximated by equation
5-10. Equation 5-13 represents the exact distance traveled and also represents the exact area
under the curve on figure 7 between tA and tB.
Examples:
1.
Give the physical interpretation of the following equation relating
the work, W , done when a force, F, moves a body from position
x1 to x2.
MA-05
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