double integral

The limit of a double Riemann sum $\sum f(x^*,y^*)\,\Delta A$ over a region — signed volume under a surface $z=f(x,y)$ .

For a partition of a rectangle $R$ into subrectangles $R_{ij}$ , each of area $\Delta A$ , and a sample point in each, the double Riemann sum is $\sum_{i}\sum_{j} f(x_{ij}^*,y_{ij}^*)\,\Delta A$ . If $f$ is continuous on $R$ , these sums converge to the same number as the partition refines, regardless of sample points — that limit is the double integral $\iint_R f(x,y)\,dA$ . When $f(x,y)\geq0$ on $R$ , this number is exactly the volume between $R$ and the surface $z=f(x,y)$ ; when $f$ changes sign, the integral is signed volume, the same way a signed single integral subtracts area below the $x$ -axis.

Examples

Estimating $\iint_R (6-x-y)\,dA$ over $R=[0,2]\times[0,2]$ with a $2\times2$ midpoint grid gives $16$ , matching the exact value.