# 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}

=> ...

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- BEng, Allahabad University, India
- MSc , Pune University, India
- PhD (IP), Pune University, India

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