Why is the volume of Torricelli's (Gabriel’s) trumpet finite?

Question

I’m aware that Torricelli’s trumpet has a infinite surface area, but why does it have a finite volume?

Answer 1

Because $\int_1^{\infty} dx/x^2$ converges while $\int_1^{\infty} dx/x$ diverges.

Answer 2

(...) but why does it have a finite volume?

Because the (improper) integral that we take as a model or definition of the (generalized) volume of this "unbounded object" is: $$\pi\int_1^{+\infty} \frac{1}{x^2}\,\mbox{d}x$$ and this integral is convergent; it has value $\pi$: $$\pi\int_1^{+\infty} \frac{1}{x^2}\,\mbox{d}x = \pi\lim_{k \to +\infty}\int_1^k \frac{1}{x^2}\,\mbox{d}x = \pi\lim_{k \to +\infty} \left[ -\frac{1}{x} \right]_1^k = \pi$$

Answer 3

It is simple, imagine you have a cylinder with a constant volume, now if you reduce its radius and increase the height, it becomes a slimmer cylinder with a constant volume and larger surface area than the original cylinder. If you keep doing that you will get a cylinder with a finite volume and infinite surface area. This is what exactly happens to Torricelli’s trumpet.