TITLE: Intuition behind this interesting calculus result?
QUESTION [51 upvotes]: So, $$\lim_{A\rightarrow \infty}\int^A_1 \frac{\ln x}{x} dx = \infty$$ All good.
But, if you take the volume of revolution of the curve, rotated about the $x$ axis $2\pi$ radians, from $1$ to $A$, the volume remains finite as $A \rightarrow \infty$ (it approaches $2\pi$):
$$\lim_{A\rightarrow \infty}\pi\int^A_1 \left(\frac{\ln x}{x}\right)^2 dx = 2\pi$$
Pretty cool result, can anyone help me get my head round this intuitively? You'd assume an infinite area rotated around the x axis would produce an infinite volume... right?

REPLY [3 votes]: Note that, for instance,  $$\int_1^\infty\frac1 {x^2}\,dx <\infty$$  so the area below the curve is finite, while the length of the curve is infinite.
What is happening is simply that the length does not increase as fast as the height shrinks. 
Similarly, in your example, the area does not increase as fast as the diameter decreases.