The average value of a function $f(x_0,y_0)$ over a given region $R$ is given by taking the volume of the resulting solid and dividing by the area over which the volume is defined. In notation, $$f(x_0,y_0) = \frac{\int \int _{R} f(x,y)dA}{\int \int _{R} dA}$$ Geometrically, this occurs when the volume of the solid equals the volume of one box... $$V = \int \int _{R} f(x,y)dA = f(x_0,y_0) \int \int _{R} dA = height \times base$$ Adjust values of $f(x_0,y_0)$ above to determine the correct height so that the volume of the region under the surface equals the volume of the single rectangular box. The height of this box will be the average value of $f$.

Average of $f(x,y) =$ .

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