5.3 Double integrals as limits

Over the small rectangle $R_{ij}$ that has South-West vertex $(x_{i},y_{j})$, we have a column of sand with height approximately $f(x_{i},y_{j})$ and base of area $hk$, so the volume of this column is approximately $f(x_{i},y_{j})hk$. Hence the total volume of the sand is approximately $\sum_{i,j}f(x_{i},y_{j})hk$. It can be shown that, as $h,k\rightarrow 0+$, these sums converge; so we define

to be the double integral of $f$ over $R$.

Later we shall calculate integrals over other regions, using the fundamental idea that the double integral of $f$ over $R$ is the volume under the surface $\{z=f(x,y):(x,y)\in R\}$ above the region $R$.