16.1 | The Definite Integral of a Function of Two Variables

The double integral of $$f$$ over the rectangle $$R$$ is
$$\iint_R f(x,y) \; dA = \lim_{m,n \rightarrow \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*,y_{ij}^*) \Delta A$$
If the limit exists, $$f$$ is called integrable. The expression inside the limit is called a double Riemann sum.
If $$f(x,y) \geq 0$$, then the volume $$V$$ of the solid that lies above the rectangle $$R$$ and below the surface $$z=f(x,y)$$ is
$$V = \iint_R f(x,y) \; dA$$
The average value of a function $$f$$ of two variables defined on a rectangle $$R$$ is given by
$$f_{\text{avg}} = \frac{1}{A(R)} \iint_R f(x,y) \; dA$$
where $$A(R)$$ is the area of $$R$$.