If
then
To compute the iterated integral, first find for which
so that
Then find for which
and
we find
Putting these together, we have
Let
.
Compute
Let
.
Compute
If the region is defined by a constant bound on
with the
bound on
varying as a function of
, then one may compute a
double integral over
as an iterated integral where one first
integrates with respect to
and then with respect to
.
Let
.
Compute
If the region is a rectangle with horizontal and vertical sides
relative to the coordinate axes, the one may compute a double integral
over
as an iterated integral by first integrating with respect to
and
then with respect to
or vice versa.
Let
.
Compute