Elementary numerical methods for double integrals

Abstract: Approximations to the integral $\int_a^b\int_c^d f(x,y)\,dy\,dx$ are obtained under the assumption that the partial derivatives of the integrand are in an $L^p$ space, for some $1\leq p\leq\infty$. We assume ${\lVert f_{xy}\rVert}_p$ is bounded (integration over $[a,b]\times[c,d]$), assume ${\lVert f_x(\cdot,c)\rVert}_p$ and ${\lVert f_x(\cdot,d)\rVert}_p$ are bounded (integration over $[a,b]$), and assume ${\lVert f_y(a,\cdot)\rVert}_p$ and ${\lVert f_y(b,\cdot)\rVert}_p$ are bounded (integration over $[c,d]$). The methods are elementary, using only integration by parts and H\"older'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.