riemann integration. introduction partitions norm of a partition
TRANSCRIPT
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RIEMANN INTEGRATION
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INTRODUCTION
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PARTITIONSPARTITIONS
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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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RIEMANN INTEGRAL
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CONDITION OF INTEGRABILITY
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Assignmentf(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 xm dx,m≠-1 on [a,b]?
Prove the condition of integrability?
Give an example of a function which is bounded but not integrable?