Math Help - 1/sqrt(x) - finite area, infinite volume?

1. 1/sqrt(x) - finite area, infinite volume?

If I revolve a finite area about an axis, do I have to get a finite volume?

I know about Gabriel's Horn / Torcelli's trumpet (same thing right?) which have an infinite surface area, but a finite volume. However, this problem by my teacher seems to be a bit different, I have a finite area about an axis and I have to see if I get an infinite volume. He told me to use

f(x) = 1/(sqrt(x))

But wouldn't this just be like Gab's Horn/TT and have a finite volume? I have to try to get an inifite volume. Or am I reading the question incorrectly? That's the exact wording in bold.

2. 1 more thing, I calculuated the volume by rotation for f(x) = 1/(sqrt(x)) and it turned out to be infinite.

I had ln(infinity) minus ln(0)

That's infinity minus undefined. So there is no finite volume, methinks. So what's this part abotu a "finite area" about an axis? I thought 1/sqrt(X) would give me an infinite area.