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Calculus : Partitions and Riemann Integral

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Let f be the function:
F(x)
1/4 x=o,
x 0<x<1
3/4 x=1

Using standard partition Pn (0,1) where n greater or equal to 4

L(f, Pn) = 2n(squared) -3n+4 all divided by 4n(squared)

U(f,Pn) = 2n(squared) +3n+4 all divided by 4n(squared)

and deduce that f is intergrable on (0,1) and evaluate
(intergral sign with 1 at top and 0 at bottom) of f

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Solution Summary

A function is proven to be integrable over a given range.

Solution Preview

Since,
dL/dn =0=> n = 8/3 and d^2L/dn^2 = +ve
=> L has minima at n =8/3 < 4 => increasing fn for n>= 4
therefore,
suprema(L) = lim(n->infinity) L = lim(n->inf) {2/4 -3/4n +4/4n^2}
=> ...

Solution provided by:
Amrit Lal Ahuja, PhD (IP)
Education
  • BEng, Allahabad University, India
  • MSc , Pune University, India
  • PhD (IP), Pune University, India
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