Integrating over the Intersection of Convex Regions Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)?  
The concrete problem I have, is to integrate over the intersection of two ellipses and a parallelogram, all centered at the origin (the ellipses and the parallelogram also depend on additional parameters).
While it would be possible to calculate the intersection, it lacks elegance and mathematical beauty and I wonder whether there is something analogous to lagrange multipliers.  
edit
In view of the answers and comments, I see the need of stressing, that calculating the intersection of regions refers to determining the intersection of the respective point sets of the given convex regions and not to determining the measure (i.e. area or volume) of the intersection.
The motivation is to avoid tedious calculations of the intersection of region boundaries in order to determine the region-intersection via a description of its boundary.     
 A: rethinking the suggestion of Joonas Ilmavirta, to use characteristic functions of the convex sets, the following recipe seems to yield a method to calculate the integral of a function over the outcome of some set theoretic combination of a (finite) collection of compact regions of the respective euclideans space:  


*

*chose an orthonormal basis of at least all functions defined over an $n$-dimensional interval (which could e.g. mean a restriction to periodic functions in case of representing functions via their fourier series), which contains the union of the regions.  

*represent the characteristic function of each region in that basis.  

*model intersections of regions as products of the respective characteristic functions ,i.e. $\chi_{A\cap B} = \chi_A\ .\chi_B$ and unions according to the laws for combining probabilities, i.e. $\chi_{A\cup B} = \chi_A+\chi_B-\chi_A\ .\chi_B$, which in the special case of two regions means subtracting the product of the respective characteristic functions from their sum.  

*express the set theoretic combination of the regions via multiplications, additions and subtractions of the characteristic functions in the same manner as is done for calculating combined probabilities and multiply the outcome with the function to be integrated.  
That method would also yield the area of arbitrary set theoretic combinations of arbitrary finite collections of compact regions of euclidean n-space.
A: One could work backwards from the answer to determine the area of intersection.
Any method would imply you had immediate knowledge of the area.
So it cannot be sidestepped.
The way to proceed is to approximate the area as well as possible. Even just the area of intersection of three random circles is practically impossible, see this question.
