Challenge Level

The $y$-axis is an asymptote for the following curves:

$$y =- \frac{1}{x} \quad\quad y= -\frac{1}{x^2} \quad\quad y^2= \frac{1}{x^3}\quad\quad y = \ln(x)-1$$

Imagine rotating the $x> 0, y< -1$ regions of these curves about the $y$-axis to form a set of hollow vessels. Which vessels are of finite volume?

Numerical extension questions

- Imagine that someone wishes accurately to engineer flasks for which the interior surfaces are given by these equations. They are to be truncated with a sealed base of radius $1$ micron, and a top opening of radius $1$cm. What would be their storage capacites and sizes? Are the resulting sizes such that you could envisage good approximations to such flasks being practically possible to make?
- Imagine that similar vessels are made with truncated and open tops (of radius $1$cm) and bottoms.They are to be designed so that they do not leak when filled with water. How long would such flasks need to be? (Assume that the diameter of a water molecule is $3$nm)
- Imagine that such flasks are made with open narrow ends of diameter $4$nm. Water is forcibly pumped into the flasks at a rate of $1$cm$^3$ s$^{-1}$, to create a jet of water consisting of a single water molecule. How fast would such jets emerge?