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.