# Calculus : Partitions and Riemann Integral

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

https://brainmass.com/math/derivatives/calculus-partitions-riemann-integral-7927

#### 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 Summary

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

$2.19