The area under the curve need not be the same though. There are many things that can change the area under the curve. A vaccine is an obvious one, once one hits the area doesn't change, so the less there is the better. (ecconmics can debate how much we should have reopened after that) the virus could also mutate and affect everyone again thus increasing the area.

I have never been able flattening the curve. If people would just stay home for a month and take precautions this would be over as it wouldn't spread and those who have it get over it. The details of living like that are tricky but too many people didn't try.