In this paper, we present an efficient approach to compute the integral of monomials and polynomials over polyhedra and regions defined by parametric curved boundary surfaces. We use Euler's theorem for homogeneous functions in combination with Stokes's theorem to reduce the integration of a monomial over a three-dimensional solid to its boundary. If the solid is a polytope, through a recursive application of these theorems, the integral is further reduced to just the evaluation of the monomial and its derivatives at the vertices of the polytope. The present approach is simpler than existing techniques that rely on repeated use of the divergence theorem, which require the antiderivative of the monomials and the projection of these functions onto hyperplanes. For convex and nonconvex polytopes, our approach does not introduce any approximation for the integration of monomials. For curved solid regions bounded by surfaces that admit a parameterization, the same approach yields simplified formulas to compute the integral of any homogeneous function, including monomials. For surfaces parameterized by polynomial surfaces (such as Bézier surface triangles and B-spline patches), the method yields machine-precision accuracy for the volumetric integration of monomials with an appropriate quadrature rule. Numerical examples over regions bounded by polynomial surfaces and rational surfaces are presented to establish the accuracy and efficiency of the method.

在本文中，我们提出了一种有效的方法来计算多面体和由参数化弯曲边界面定义的区域上的多项式和多项式的积分。我们将Euler定理用于齐次函数，并与Stokes定理结合使用，以减少一维多项式在其三维实体上的积分。如果实体是多面体，则通过递归应用这些定理，可将积分进一步简化为仅对单面式及其在多面体顶点处的导数求值。本方法比依赖于重复使用发散定理的现有技术更为简单，后者需要单项式的反导并将这些函数投影到超平面上。对于凸和非凸多面体，我们的方法没有为单项式积分的集成引入任何近似值。对于以允许参数化的曲面为边界的弯曲实体区域，相同的方法会得出简化的公式，以计算任何齐次函数（包括单项式）的积分。对于由多项式曲面参数化的曲面（例如Bézier曲面三角形和B样条斑块），对于具有合适的正交规则的单项式体积积分，该方法可产生机器精度的精度。给出了以多项式曲面和有理曲面为边界的区域上的数值示例，以建立该方法的准确性和效率。同样的方法产生简化的公式来计算任何齐次函数的积分，包括单项式。对于由多项式曲面参数化的曲面（例如Bézier曲面三角形和B样条斑块），对于具有合适的正交规则的单项式体积积分，该方法可产生机器精度的精度。给出了以多项式曲面和有理曲面为边界的区域上的数值示例，以建立该方法的准确性和效率。同样的方法产生简化的公式来计算任何齐次函数的积分，包括单项式。对于由多项式曲面参数化的曲面（例如Bézier曲面三角形和B样条斑块），对于具有合适的正交规则的单项式体积积分，该方法可产生机器精度的精度。给出了以多项式曲面和有理曲面为边界的区域上的数值示例，以建立该方法的准确性和效率。