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 \(R\).

E 16.1 Exercises