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The integrals over B and T will be positive. Keeping fixed,
is strictly increasing over D as
increases, so the integrals over
(i.e. the bottom/top left quadrants of D) is negative but the integrals over
are *more* positive.
The integrals over R and L are zero. If we take , then
, which is to say
is symmetric across the
-axis. For the same reason, the integral over all of D is also zero.