Correct area calculation. I would now employ the
trapezoidal rule to formulate the (exact) area under a sequence of joined piecewise linear segments. Question: are the x-coordinates evenly spaced? If so, you can use the usual composite trapezoidal rule. If not, you have to use the formula for non-uniform intervals, as also provided in the link.
In answer to your Post # 22, I would say yes: you must check every permutation, essentially. There might be a way to short-cut this, if you can see some theorem or other that you can prove. For example: take a close look at the formula for uniform intervals. Do you notice anything about it?