Each table is 3 seats long and 1 seat wide. If you push 3 tables together, end to end, they become 9 seats long, and remains 1 seat wide. If you add up the perimeter, it comes out to 9+9+3+3, which is 24.
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.