Gabriel's horn, calculus Integration

Question

Gabriel's horn is formed by revolving the curve $y=1/x$ for $x\in [1,\infty)$ about the $x$-axis.

Find the volume inside Gabriel's horn. I have the answer but I can't seem to get it right. Can someone explain please?

Answer 1

What you should do for finding the result is to evaluate the following integral $$\int_1^\infty \pi y^2 dx$$ in which $y=\frac{1}{x}$. You can draw a disc as you see in fig below. We build this disk on $x$- axes, so the volume of it is the volume of colored cylinder. What is that volume? It is $\pi r^2 h$. What is $r$ and what is $h$? indeed, $r$ is $y$ and $h$ is $dx$.