  We provide a new approach for computing integrals over hypersurfaces in the
level set framework. The method is based on the discretization (via simple
Riemann sums) of the classical formulation used in the level set framework,
with the choice of specific kernels supported on a tubular neighborhood around
the interface to approximate the Dirac delta function. The novelty lies in the
choice of kernels, specifically its number of vanishing moments, which enables
accurate computations of integrals over a class of closed, continuous,
piecewise smooth, curves or surfaces; e.g. curves in two dimensions that
contain finite number of corners. We prove that for smooth interfaces, if the
kernel has enough vanishing moments (related to the dimension of the embedding
space), the analytical integral formulation coincides exactly with the integral
one wishes to calculate. For curves with corners and cusps, the formulation is
not exact but we provide an analytical result relating the severity of the
corner or cusp with the width of the tubular neighborhood. We show numerical
examples demonstrating the capability of the approach, especially for
integrating over piecewise smooth interfaces and for computing integrals where
the integrand is only Lipschitz continuous or has an integrable singularity.
