Intuition behind this interesting calculus result?

Solution 1:

Yes, this was considered a pretty strange phenomenon when Torricelli first constructed such an example in 1643. (Torricelli himself found it so incredible that he offered two different proofs that the volume of his shape was finite, the better to convince himself and the reader).

I think there are two main mental hurdles that one must conquer in order to shake the impression that this is paradoxical:

• An infinite sequence of positive numbers can have finite sum.

• Volumes of revolution with the same finite cross-sectional area can have as large or small volume as you want.

The first of these has occupied minds since Zeno's paradoxes. Most of us are eventually able to come to sort of peace with it, if only by bludgeoning ourselves with the many examples of it.

For the second, just compare the volumes of cylinders you get by rotating a rectangle of unit area around one of its sides. The sides of the rectangle is $a$ by $\frac1a$ for some $a$, and calculating its volume we get $\pi a^2\cdot\frac1a = \pi a$ which we can make anything we want just by choosing $a$ appropriately. What's going on is that a small piece of area close to the rotation axis contributes less volume than the same area longer away, so by choosing a long thin rectangle that hugs the axis tightly, we can make the volume as small as we want to.

Putting these two together: Choose a convergent series of positive numbers $$ b_1 + b_2 + b_3 + \cdots = c < \infty$$ and for each $b_i$ pick the cylinder with volume $b_i$ and cross-sectional area $1$. Stack all of these cylinders after one another, and you get an infinitely long volume of revolution with infinite cross-section but finite volume, namely $c$.

This is perhaps not as beautiful as your nice smooth $\frac{\log x}{x}$, but is arguably easier to wrap one's head around because one needs only deal with discrete series rather than continuous integrals.

Solution 2:

Yes, it is an example that shows that we cannot assume that ''an infinite area rotated around the x axis would produce an infinite volume''.

Intuitively this is because the area goes to infinity because $f(x)=\frac{\ln x}{x}$ goes to $0$ too slowly, but the volume $f^2(x)=\left(\frac{\ln x}{x}\right)^2$ goes to $0$ more quickly.