Il logo di batmath
previous page | next page

Some properties of the Riemann integral

The same properties already considered for the integral of step functions remain unchanged for the Riemann integral. Their proof is not difficult: in some cases it can be obtained by simple generalisation of the same property for the integral of step functions, in other cases it is a simple consequence of the Vitali-Lebesgue theorem.


The linearity property can obviously be generalized to any finite number of functions as follows: if f1, ... , fn are functions integrable on the same interval [a,b], then c1f1 + ... + cnfn is also integrable and img, for any choice of the constants c1, ... , cn

Observe that if two functions are not integrable, their sum may be integrable: it's enough to take a non integrable function and the opposite one, so the sum is zero. The same holds for the product and the quotient of two non integrable functions.

The comparison property has an important consequence: setting f(x)=0 you obtain img, if g(x)0: the Riemann integral of a positive function is positive. The converse is not true: if the integral of a function is positive, the function is not necessarily positive. A simple example can be given using the function signum (as this is a step function its integral is easy to calculate): we have img, but the function is not positive in the interval [-1,2].

As far as the absolute value property is concerned observe that the converse of the property is not true: the absolute value of a function may be integrable, while the function is not. Consider for example img, whose absolute value is a constant function.

previous page | next page
first published on january 07 2003 - last updated on december 20 2003