Unformatted text preview: t is easy to see that the energy E, as a function of displacement, will
energy
be infinite. And hence: E = ∫ m.a.ds will not CONVERGE.
m.a.ds
Here we implicitly assumed that the length of the pendulum was infinite. What if the
pendulum was of finite length ? Does it mean that the pendulum will swing all the way
round in a circular orbit ?
What if the ball bounces indefinitely over infinite time interval [0, +∞) with gain in
energy on each bounce ? Will the ball disappear both in TIME and SPACE ?
It is of fundamental importance to see the relationship between the abstract calculation
on the one hand and the physical meaning ( the reality  what is happening in nature)
on the other hand.
With all this mind we now give informal definition of INTEGRABLE.
1. f(x) is defined over [a, b] (that is to say SINGLE VALUED) except at the points of
discontinuity. And f(x) has at most COUNTABLY many discontinuities over [a, b]. So between each pair of consecutive such points of discontinuity we have a
subinterval.
2. An each such point of discontinuity f(x) has 2 definite values.
discontinuity 3. continuous summation
Over each of the subintervals in [a, b] the continuous summation
contin
must be something FINITE, say : S0 , S1 , S2 , . . . , Sn , . . . . 4. The combined discrete summation S = { S0 + S1 + S2 + . . . + Sn + . . . }
discrete ∫f(x)dx over all the COUNTABLY many subintervals must also be something FINITE
b so that ∫ f (x)dx CONVERGES.
a Then we say f(x) is INTEGRABLE. 174 A word on CONVERGES
The fundamental requirement in integration is that the continuous summation
integration
continuous
CONVERGES. Let us compare two examples of discrete summation and continuous
discrete
continuous
summation .
The discrete summation
discrete
+1 ∞ 1
Σ /n does NOT CONVERGE.
n=0 The integral ∫ { 1/x} dx is a continuous summation over the finite interval [0, +1].
continuous summation
contin
0
But the summation does not CONVERGE. So it is not INTEGRABLE.
+∞
f(x) = 1 x
/ 2
1 ½
0 −∞ ½1 2 ... +∞ −1
−2 −∞
We can see from the diagram that the area under the curve f(x) = 1 x over the finite
/
inter...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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