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