Unformatted text preview: sion of CHANGE. 169 33. I N T E G R A B L E
We have used the word INTEGRABLE without giving a formal definition. Here we give
an informal definition with a geometrical explanation. We also present an intepretation
from a Physics point of view with the focus on position and energy .
position
energy
d F(x)
Let us start with a strict definition. Let f(x) = dx .
only
single valued,
Then f(x) is INTEGRABLE iff and only if F(x) is WELLBEHAVED  single valued,
i
sing
continuous and differentiable . The emphasis here is on the differentiable
differentiable
differentiable
property. single valued
ontinuous
uous.
We may be less strict by letting F(x) be just single valued and continuous. In this
sing
case it is possible that f(x) will be discontinuous at some instants as in the example
discontinuous
below. At such instants where f(x) is discontinuous we require f(x) to have two
discontinuous
discontin
definite values.
f(x) = F(x) = x
+1 d F(x)
dx 765432
7654321
7654321
7654321
7654321
1
0987654321
7
0987654321654321
0987654321
0987654321
0987654321
0987654321 a 0 b x −1
0
x
We see that f(x) is defined over the subintervals [a, 0 −) and (0 +, b] . Notice that f(x)
is discontinuous at 0 but it has two definite values.
discontinuous
f(0 −) = −1 and f(0 +) = +1
With this type of discontinuity we may still integrate f(x) over [a, b] .
b 0− b ∫ f (x)dx = ∫ f(x) dx + ∫ f(x)dx
a 0+ a 170 continuous summation
With the understanding that ∫ is a continuous summation , we want the sum to
contin
be FINITE or CONVERGE to some definite value. In this case we can see that both
0− b ∫ f(x) dx = S 1 = − a is something finite, and ∫ f(x) dx = S 2 = + b is
0+ a something finite. Also their sum does add up to a definite value .
0− b { ∫ f(x) dx } + { ∫+f(x)dx } = S1 + S 2 = − a + b
a 0 integ
We state 4 basic conditions for f(x) to be integr able .
inte
1. f(x) is defined over [a, b] (that is to say SINGLE VALUED) except at the points
of discontinuity. And f(x) has only FINITELY many n discontinuities over [a, b].
2....
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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