Section 15.1: Double Integrals over rectangles

Recall from one variable calculs that the average value of a functio \(f\) of one variable defined on an interval \([a, b]\) is \[ f_{ave} = \frac{1}{b-a} \int_a^b f(x) dx \] In a similar fashion we define the average value of a function \(f\) of two variables defined on a rectangle \(R\) to be \[ f_{ave} = \frac{1}{A(R)} \iint_R f(x, y) dA \] where \(A(R)\) is the area of \(R\). If \(f(x, y) \ge 0\), the equation \[ A(R) \times f_{ave} = \iint_R f(x, y) dA \] says that the box with base \(R\) and height \(f_{ave}\) has the same volume as the solid that lies under the graph of \(f\).