Double Integrals
Table of Contents
1. Introduction
Consider taking the volume of a surface \( S = f(x, y) \) over some rectangular region \( R \) of the \( (x, y) \) plane.
We can approximate the volume by partitioning the rectangular region \( R \) into an \( m \) by \( n \) grid to obtain \( m n \) lots of rectangles.
The double integral is defined as:
\[ \iint_R f(x, \, y) dA = \lim_{m,n \rightarrow \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, \, y_{ij}^*)\Delta A \]
This converges as any continuous function is Riemann integrable.