This is a classic conundrum, and it's just the way the math works out.
Standard example: Revolve y = 1/x, domain [1, infinity) about the x-axis. The surface area is infinite, but the volume is finite.
How is that possible? I dunno. It tells me to beware my biological/visceral-intuitions that combine "what paint can do" with notions of infinities and convergences. Instead, I trust more to my math-intuitions (limited though they be) when it comes to infinities and convergences. If you've some other perspective on how this seeming paradox should be understood, I'd be curious to hear it.