Computing Integrals over Polynomially Defined Planar Regions and Curves

Michael Wester, Yuzita Yaacob, Stanly Steinberg 

Abstract: We use the cylindrical algebraic decomposition algorithm
implemented in Mathematica to produce algorithms to analytically
compute integrals over polynomially defined regions in the plane.
We also produce an algorithm to compute integrals over the boundary
of such regions and then implement the two dimensional version of
the Green's theorem. The resulting software can solve many of the
two dimensional integration problems in calculus textbook.