Area of enclosed smooth curve is always irrational for rational dimensions

Question

I have the intuition that smooth curves that are enclosed cannot be possible in real world , so it must be the case that either the dimensions(parameters) that define the curve ( like the radius of circle) are irrational or for real dimensions the area of the curve is irrational like for example for circle () and ellipse (). Is there any such proof (for or against) for this conjecture for generic smooth enclosed curve ?.

Answer 1

The area enclosed between a parabola $y=x^2$, $x$-axis, and vertical lines $x=p$ and $x=q$ for any rational numbers $p<q$ is $\int_p^q x^2=\frac {x^3}3\big|^q_p=\frac {q^3}3-\frac {p^3}3$, that is rational. The similar situation holds for any (necessarily smooth) curve of the form $y=P(x)$, where $P$ is a polynomial with rational coefficients. If you wish to have a rational area completely enclosed by a smooth curve you can glue together several such parts as above. For instance, we can glue together four parabolic pieces $y=\pm \left(x^2-\frac 34\right)$, $x=\left[-1/2, 1/2\right]$ and $x=\pm \left(y^2-\frac 34\right)$, $y=\left[-1/2, 1/2\right]$.

Pythagoreans taught that natural numbers create the world base, so only their fractions, rational numbers are possible in the real word. Moreover, there is a legend, similar to ancient Greek drama telling when one of them, Hipatius, found that the length of a diagonal of a square with unit sides is irrational, he was thrown down a ship into Mediterranean sea by other pythagoreans, and they decided to forget this theorem and never discover it again on death penalty.

Conic sections: circles, ellipses, parabolas are used in the classical mechanical world model. In particular, as trajectories of falling bodies or planets rotating around the Sun. Moreover, as far as I know, both Johannes Kepler and Isaac Newton who discovered and derived these motion laws, believed that they revealed God’s Universe construction plan.