$\frac{π}{2}$ is a trascendental number, given by the integral:
$$\frac{π}{2}=\int_{-1}^1\sqrt{1-x^2}dx$$
The integrand is also the curve of a semiball of the form:
$$||(x,y)||_2=1$$
Induced by the 2-norm $||\cdot||_2:\mathbb{R^2\rightarrow R_+}$ where $y$ has been parametrized in terms of $x$.
Similarly, we can get an integral representation of the set of real numbers $S=\{υ_1,υ_2,υ_3...\}$; $υ_\infty\notin S$, by defining: $$υ_k=\int_{-1}^1B_k(x)dx$$ Where $B_k(x)$ is the semiball obtained by parametrizing the curve $$B_k\equiv||(x,y)||_{2k}=1$$ Formed by the $p$-norm $||\cdot||_p:\mathbb{R^2\rightarrow R_+}$ in terms of $x$. In other words: $$B_k(x)=\sqrt[2k]{1-x^{2k}}$$ (Evidently, in this setting, $υ_1=\frac{π}{2}$). My question is then: Is every element of the set $S$ a trascendental number? Any insight or information about this will be appreciated.
