Spin a curve about an axis and it sweeps out a solid. Slice that solid perpendicular to the axis and every slice is a disk of radius f(x): stack them and the volume is ∫ π f(x)² dx.
Cut the same solid the other way — into thin nested tubes parallel to the axis — and each is a shell of area 2πy. Two knives, one solid, one answer.
A Riemann sum with n slices is only an estimate; as n grows the staircase melts into the true integral. That limit is the definite integral.
Torricelli's trumpet — y = 1/x spun out to infinity — has a finite volume of exactly π, yet its surface never stops growing. You could fill it with paint but never coat it.