# Elementary numerical methods for double integrals

## TL;DR

This paper develops elementary numerical methods for approximating double integrals, providing error estimates based on the boundedness of certain partial derivatives in an $L^p$ space, using integration by parts and Hölder's inequality.

## Contribution

It introduces new error bounds for classical quadrature rules for double integrals under $L^p$ derivative assumptions, using simple analytical techniques.

## Key findings

- Error estimates for trapezoidal and midpoint rules derived
- Methods applicable under minimal smoothness assumptions
- Elementary approach using integration by parts and Hölder's inequality

## 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.