Proving and calculation an integral from the definition

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• May 24th 2011, 08:15 AM
moses
Proving and calculation an integral from the definition
Hello

I have the following problem: given that f : [a,b] > R is integrable, prove that
g = f/2 is integrable and find the integral.

I'm learning Darboux integrals, so my instinct is to try and calculate the upper and lower sums of g and show that for all epsilon, there exists a partition of [a,b] such that the difference between them is less than epsilon...but I'm not sure how I can break apart the difference in order to use the fact that f is integrable...i'm very confused.
• May 24th 2011, 09:59 AM
HallsofIvy
It is not necessary to look at upper and lower sums. Given any specific partition, S, of [a, b], and any set of x values, \$\displaystyle {x_i}\$, consisting of one point in each subset, call that sum A_S. Because f is integrable, the limit, using any partition and any choice of x values, exist and is equal to the integral of f over [a,b]. Using the same partition and x values for g, gives (1/2)A_S, which clearly also converges to 1/2 the integral of f over [a,b].