Conceptual Question of Area and Volume

[wwshr87]

Why the area under the curve of the function y=1/x form 1 to inf is infinite. But if we take this area and revolve it around the y-axis we obtain a volume of pi ?

 

[Raskolnikov]

Because the volume of this rotated solid converges while its surface area diverges. Just because Volume is "one-dimension more" than Area doesn't mean it's "larger." For example, a 0.1m by 0.1m square has an area of 0.01 m^2, but a 0.1 x 0.1 x 0.1 cube has a volume of 0.001 m^3. In your problem, the V_n simple converges quicker than A_n.

Your topic heading said "conceptual question," but here's the math explanation as well if you're curious:

$$A = \int_1^{\infty} \frac{dx}{x} \ = \ {lim}_{b \rightarrow \ \infty} ( \ ln|x|_1^{b} \ ) = b - 0 = b = \infty.$$

On the other hand,

$$V = \int_1^{\infty} \pi(\frac{1}{x})^2 * dx = \pi \int_1^{\infty}\frac{dx}{x^2} = \pi * {lim}_{b \rightarrow \ \infty} [ \ -x^{-1}|_1^{b} \ ] = 0 - (-\pi) = \pi.$$

 

[wwshr87]

Well that makes perfect sense. I knew about the mathematical explanation but was looking for the other explanation you provided. Thank you very much sir.