[NicolasM]

So, we have a hydrogen atom. What is the probability of an electron being found in a spherical shell with 0,01Å width and a radious of 0,35Å?

By approximation, we can sonsider the value of ψ (1s) stable within the shell, and thus the chance is ψ² x Volume. Right?
Also, by considering 0,35Å the median radious, can I express the probability as _0,345∫^0,355 ₀∫^2π ₀∫^2π ψ² r² sineφ dr dφ dθ ? Would that give a more accurate solution?

[Corribus]

You just need to integrate the radial wavefunction (probability function) over the distances specified.

[Enthalpy]

ψ²x Volume approximately : yes.

The integral would be more precise... provided you write it properly. The bounds are false.

You can write the integral more simply from the beginning, since ψ depends only on r. Integrate over r, with a volume element 4πr²*dr
(area of sphere * dr).

Of course, this all supposes that Ψ is properly scaled, with |Ψ|² summing to 1 over space. And |Ψ|² because ψ is a complex function.