oh I just came up with a terribly cursed thing that I'm not sure is mathematically valid?
the integral of sin(x) from 0 to infinity is actually zero.
if we take the integral from 0 to 2*pi, it is zero. by symmetry the integral of every such period is zero. if we take the sum of every such period's integral, which is infinitely long, we get an infinite sum of zero, which is zero.
this, i think, actually brings up an interesting philosophical point about what infinity even means in a thing like this.
i personally don't think completed/definite infinity is even a real thing, but instead it's a process of building up that never terminates.
and that viewpoint shows in this sketch of maybe-proof. but if you do think there's a definite infinity, then this makes no sense. what's sine of infinity? this doesn't converge to any limit.
@KitRedgrave The problem is that unless you specifically infinitely move in multiples of 2π, it still doesn't pan out. Sure, we can keep moving on and on but it will just move up and down, constantly fluctuating. You can never define a value if you're just constantly moving across sine. To define a value in this case would inherently be to define a stopping point.
