Elementary numerical methods for double integrals

Abstract

Approximations to the integral ${\int }_a^b{\int }_c^{d}f(x,y)dydx$ are obtained under the assumption that the partial derivatives of the integrand are in an $L^p$ space, for some $1\le p\le \infty$. We assume ${\Vert f_{xy}\Vert }_p$ is bounded (integration over $[a,b]\times [c,d]$), assume ${\Vert f_x(\cdot ,c)\Vert }_p$ and ${\Vert f_x(\cdot ,d)\Vert }_p$ are bounded (integration over $[a,b]$), and assume ${\Vert f_y(a,\cdot )\Vert }_p$ and ${\Vert f_y(b,\cdot )\Vert }_p$ are bounded (integration over $[c,d]$). The methods are elementary, using only integration by parts and Hölder's inequality. Versions of the trapezoidal rule, composite trapezoidal rule, midpoint rule and composite midpoint rule are given, with error estimates in terms of the above norms.

To appear in Minnesota Journal of Undergraduate Mathematics

References in corpus (1)

• Simple derivation of basic quadrature formulas