I'm trying to figure out if there is a solution to finding the volume of the shape that is formed by rotating $f(x) = \frac 1x$ about the y-axis. Not the x-axis as in Gabriel's horn, but the same problem about the y-axis on the interval $x \ge 1$.
Using the shell method and improper integrals, I end up with $\pi$($\infty$ - 1) where b $\to \infty$.
Is the volume on this infinite? If so, what is the correct way to prove that it is infinite?
Using the disc (washer) method, the integral to set up is $\int_0^1 \pi (x^2 -1) \; dy$ and you need to recall that $y = \frac{1}{x}$, so $x = \frac{1}{y}$. This integral is improper as the lower limit of integration is not in the domain of the integrand.