[READ]Unusual geometrical shapes

1)In fractal geometry there exist shapes which have INFINITE perimeter and FINITE area (Koch Snowflake). Which I find amazing, however I was wondering what if the positions were reversed? i.e. INFINITE area but FINITE perimeter. I doubt that this is true but I cannot proof it. For one thing there is no well-defined term for a "fractal" we can just give examples.

2)This question is related to the previous. There are solids called "supersolids" these are solids which have INFINITE surface area but FINITE volume (Gabriel's Horn-you can fill it with paint but not paint it with paint). Now reverse the positions, INFINITE volume but INFINITE surface area. However, this problem I think is easier because we **can** give a well-defined term to a supersolid. We can define it like this: If a function is countinous and positive for x>a, then show that if the improper integral for surface area converges if and only if the improper integral for volume converges! This might actually be true which I am going to try to prove.