Riemann Integral

Let and (with )
Let be a function.
Suppose is Riemann Integrable on
with Upper Integral and Lower Integral equal to
Then we say that the Riemann integral of is
We write:

Additionally, define:

Improper integrals

If the domain contains for some and
(note the inclusion of )
define for all :

if the limit exists.
we do a similar thing for and

If both and (where )
let and define:

(if and exist as improper integrals for all )
It can be shown that this is well defined
i.e. the result doesn’t depend on .