The area under the graph of the function $$\frac{1}{x}$$ from $$[1, \infty)$$ diverges, as demonstrated by the integral $$A = \int_{1}^{\infty} \frac{dx}{x} = \infty$$. In contrast, the volume of the solid of revolution generated by this function in the same interval converges to a finite value, calculated as $$V = \pi\ \int_{1}^{\infty} \frac{dx}{x^{2}} = \pi$$. This phenomenon, often referred to as Gabriel's Horn, illustrates a mathematical paradox first identified by Pappo di Alessandria in the fourth century.
PREREQUISITESMathematicians, calculus students, educators, and anyone interested in the paradoxes of mathematical analysis and the properties of infinite volumes.
sweatingbear said:How come the area below the graph of $$\frac 1x$$ between $$[1, \infty)$$ does not exist, but the solid of revolution below the same graph in that same interval does exist? I do not see the logic.