.
Evaluating at the limits gives
.
Integrating with respect to y gives
.
Finally evaluating at the limits gives
.
.
Integrating with respect to x gives .
Evaluating at the limits gives .
Integrating with respect to y gives .
Finally evaluating at the limits gives .
2. Evaluate 
.
The function as it stands has no elementary antiderivative with respect to x, so our only option is to reverse the order of integration.
A
sketch of the region looks something like what's shown at left, extending
from the right-opening parabola x = y2 on the left to y = 4
on the right, for slices from y = 0 below to y = 2 above.
To express the limits in the opposite order, we
go bottom-to-top and then left-to-right. The bottom curve is y = 0, and
the upper curve is 
.
Our slices run from x = 0 to x = 4.
Thus the new integral is 
.
Integrating with respect to y gives 
.
Evaluating at the limits gives 
.
This function can be integrated (using the substitution u = x2)
to leave us with 
.