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Now let us look at the situation where the ball gains

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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 WELL-BEHAVED - 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 sub-intervals [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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