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Definition: The average value or mean value of an integrable function f (x, y) over the set D is the number ZZ 1 f = avD f = f (x, y) dA . area of D D Definition: A set D in the plane is said to be connected if any two points D in can be joined by a continuous function γ(t) = x(t), y(t) : [0, 1] ⊂ R −→ D ⊂ R2 whose image lies entirely within D. Mean Value Theorem for Double Integrals If the function f (x, y) is continuous on a compact, connected set D in the xy–plane, then there exists a point (x0 , y0) in D such that ZZ f (x, y) dA = f (x0, y0) area of D . D