The question is meaningless since you can't cover the surface with paint.This statement would also be meaningless.
I'm afraid the answer is yes.
If you rotate a hyperbola x2 - y2 = 1 around the y axis, you form a body with finite volume but infinite surface area, so you can paint an infinite surface by pouring in a finite can of paint, but not by spraying or brushing it.
Economists make their living by talking about the can of paint and ignoring the brush.
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