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Published byDana Dorothy Stokes Modified over 7 years ago

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RIEMANN INTEGRATION

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INTRODUCTION

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PARTITIONS

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NORM OF A PARTITION

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REFINEMENT OF A PARTITION

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Upper and lower Riemann sums

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EXAMPLES:

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THEORMS OF UPPER AND LOWER SUM

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RIEMANN INTEGRAL

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CONDITION OF INTEGRABILITY

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Assignment f(x)=x on [0,1] where P={0,1/3,2/3,1}? If P is a partition of interval [a,b] and f is a bounded function defined on [a,b], then L(f,P) U(f,P)? f(x)=sinx on [0, /2] where P={0, /6, /3, /2}? State and prove Darboux theorem? State and prove necessary and sufficient condition of integrability? Every monotonic and bounded function is integrable?

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A continuous function on a close interval is integrable on that interval? Show that greatest integer function f(x)=[x] is integrable on [0,4] and [x]dx=6? let f be a bounded function such that the set of points of discontinuity of f on [a,b] then f is integrable on [a,b]? show that f(x)=|x| is integrable on [-1,1]

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TEST Attempt any three: State and prove Darboux theorem? Evaluate x m dx,m≠-1 on [a,b]? Prove the condition of integrability? Give an example of a function which is bounded but not integrable?

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