  The paper studies several approaches to numerical integration over a domain
defined implicitly by an indicator function such as the level set function. The
integration methods are based on subdivision, moment--fitting, local
quasi-parametrization and Monte-Carlo techniques. As an application of these
techniques, the paper addresses numerical solution of elliptic PDEs posed on
domains and manifolds defined implicitly. A higher order unfitted finite
element method (FEM) is assumed for the discretization. In such a method the
underlying mesh is not fitted to the geometry, and hence the errors of
numerical integration over curvilinear elements affect the accuracy of the
finite element solution together with approximation errors. The paper studies
the numerical complexity of the integration procedures and the performance of
unfitted FEMs which employ these tools.
