1.  the integral of a linear combination is the linear combination of the integrals,

   

If a > b then define

3.

Additivity of integration on intervals. If c is any element of [a, b], then

4.

Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and

M(b − a), it follows that

     

 

Applications in physics/technology of integrals.