The Torricelli solid/Gabriel's Horn is defined as the rotation-invariant solid delimited by a hyperbola. It appears in De solido hyperbolico acuto where Torricelli proves that it has a finite volume, despite the fact that it is infinite.
Nowadays, that paradox appears in a different guise, as the fact that the area of that surface that bounds the Torricelli solid is infinite. In simple terms, the Torricelli trumpet cannot be painted but it can be filled with paint. This is for example how the story is told in Wikipedia.
However, I do not know who made this observation regarding the area, and when.
Using integral calculus, these facts can be given relatively straightforward proofs.
Torricelli's proof that the solid has finite volume uses an approximation of the solid as a stack of “infinitesimally” small concentric cylinders. This is indeed enough to bound that volume.
However, to evaluate the area, that would not be enough and to write a proof in Torricelli's style (or in the style of the early inventors of the infinitesimal calculus), it would be necessary to consider the solid as a stack of infinitesimall truncated cones.