# Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$? [closed]

The calculation of the area of the $$\mathbb{R}^2$$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates:

$$S_{\mathbb{R}^2}=\int_0^\infty 2\pi r dr=2\pi\left(\frac{\tau^2}2+\frac1{24}\right)=\pi\tau^2+\frac\pi{12}$$

where $$\tau=\int_0^\infty dx$$.

The regularized value of this area is $$0$$. On the other hand, the area of a disk with radius $$\tau$$ (equal to the length of the real semi-axis) is $$S_c=\pi\tau^2$$, and its regularized value is $$-\frac\pi{12}$$.

Thus, $$S_{\mathbb{R}^2}-S_c=\frac\pi{12}$$. I wonder, where this area difference comes from? Does it originate from the fact that the plane should not be considered a disk of infinite radius? Or it is some glitch of integration technique?

• How on earth can it be meaningful to attach any value to this integral except $\infty$? Apr 19, 2020 at 8:39
• How do you get the integral equal to $2\pi\left(\frac{\tau^2}2+\frac1{24}\right)$? Apr 19, 2020 at 11:38
• @Wojowu the formula appears in this post of OP, I got lost before arriving to it so I cannot comment further on it. Apr 19, 2020 at 23:58
• @Anixx: You ask quite a few questions about your own theory of divergent integrals, which surely very few people are qualified to answer. Is there a good reference where your theory is developed in full rigour and detail, so that it might be possible for mathematicians other than you to answer your questions? Jan 15 at 17:22
• @BenMcKay for now, I think, there is no theory because I found that the proposed approach was non-natural definition. It seems, the Levi-Civita type of construction is more natural. On the other hand, I am looking at whether one can be merged/embedded into the other. If you want an outdated text, I can provide a reference... Jan 15 at 17:38

$$\int_0^\infty dx\cdot \int_0^\infty dx=\omega^2=2\int_0^\infty x dx$$.
So, the areas in both cases are $$\pi\omega^2$$, where $$\omega=\int_0^\infty dx$$.