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Double Integrals by Summing Values of a Cumulative Distribution Function

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Let
f
be a function, and suppose that its "cumulative distribution function"
F(x,y)=
y
∫
u=0
x
∫
v=0
f(u,v)dudv
, is known.
F(P)
is the integral of
f
over the rectangle below and to the left of
P
, and the double integral of
f
over a rectangle can be computed easily in terms of the values of
F
at the corners via:
∫∫
ABCD
f(x,y)dxdyF(A)-F(B)+F(C)-F(D)
.
Checking boxes causes a region to be shaded such that the combination
F
values at the checked corners is the integral of
f
over the shaded region.
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