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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# What is the distance travelled from t 10 secs to t 20

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Unformatted text preview: val [0, +1] is NOT FINITE. From a Geometric point of view we may define INTEGRABLE as : f(x) is INTEGRABLE ⇔ the area under f(x) is FINITE 175 ∞ The discrete summation discrete summation discr 1 Σ /2 i CONVERGES. The terms are finite in value. i=0 However, the subscript i varies over the infinite interval [0, +∞). ∞ −i also CONVERGES. Likewise the discrete summation i Σ0 e discrete = It is possible to have an integral or a continuous summaiton over an infinite continuous interval. +∞ It is now easy to see that the continuous summation ∫ continuous 0 {e−x} dx must CONVERGE even though the interval [0, +∞) is infinite. Hence f (x) = e− x is INTEGRABLE. +1 f (x) = e− x x 0 +∞ ∫ 0 {e−x} dx +∞ −x ] = [− e 0 = (− e −∞) − (− e −0 ) = 0 − (− 1) = +1 176 We may also integrate over (−∞, +∞) as in the example below. Let f(x) = { e− x for x ≥ 0 e x for x ≤ 0 +1 e− x ex 0 −x +∞ 0 x +∞ ∫ f(x) dx = ∫ f(x) d x + ∫ −∞ −∞ 0 f(x) dx = +1 +1 = +2 associati We are familiar with the associativity property in addition : (a+b)+c = a+(b+c). associativity This property holds good when the number of terms are finite. It also holds good for a discrete summation of COUNTABLY many terms if all the terms are of same discrete summation discr sign. But it breaks down when there are COUNTABLY many terms of both positive and negative sign. The alter nating series +1 −1 +1 −1 + . . . does NOT CONVERGE. alterna alternating Depending on how we do the summation we have 3 possible answers : −1, 0, and +1. So one can imagine the difficulties involved in a continuous summation with continuous UNCOUNTABLY many terms of both signs. Proving CONVERGENCE becomes very, very difficult. A more formal and rigorous treatment of INTEGRABLE dealing with the problems of CONVERGENCE - discontinuities, finiteness and associativity - is covered discontinuities finiteness associativity in A LITTILE MORE CALCULUS. 177 34. pplications Integ A pplica tions of Inte g r a tion There are numerous applications of in...
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