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 .