Q:

Integration over region given by inequality

Sometimes we need to integrate over some bounded region given by inequality/inequalities.  Consider the following simplest example of area of an ellipse (LaTeX code):
$\int_S dx dy$, where $S = \{ x^2/a^2 + y^2/b^2 \leq 1\}$.
Is it possible to do this directly in Mathematica?  If possible, how to do it? 

A:

Explicitly:
Integrate[Boole[x^2/a^2 + y^2/b^2 <= 1], {x, -a, a}, {y, -b, b}, 
          Assumptions -> {a > 0, b > 0}]

Function Boole delineates the region; the Assumptions option is needed for symbolic a and b in order to allow Mathematica actually to evaluate the integral; if a and b are numeric, that option is superfluous.