Geometric Interpretation for Definite Integrals with $\pi$ in the result What is the geometric interpretation for the following integral?
What is a nice geometric interpretation for the following integral (possibly in relationship to a circle) that emphasizes why we get the result of π in the right hand side?
$$\int^\infty_{-\infty} \frac{1}{1+x^2}dx = \pi$$
I do know, of course that the indefinite integral for the integrand is $\tan^{-1} x$.
In the answers, I'd also appreciate examples of other integrals with a geometric interpretation.

Consider the stereographic projection of the line onto the circle.
$\hspace{1cm}$
Let $x$ be the distance from the point of tangency and $r$ be the distance from the center of the circle. Using similar triangles in the right inset, we get that $\mathrm{d}x$ on the tangent line is projected onto $\frac{\mathrm{d}x}{r}$ on the piece parallel to the circle. Then, again by similar triangles in the left inset, that is reduced to $\frac{\mathrm{d}x}{r^2}$. Thus, the projection of $\mathrm{d}x$ onto the circle yields
$$
\frac{\mathrm{d}x}{r^2}=\frac{\mathrm{d}x}{1+x^2}\tag{1}
$$
Integrating $(1)$ over the entire real line will give the length of half the circle, $\pi$. That is,
$$
\int_{-\infty}^\infty\frac{\mathrm{d}x}{1+x^2}=\pi\tag{2}
$$
