What would be the average value of $r$, i.e. $\langle r\rangle$, in the $\mathrm{1s}$ orbital of $\ce{He+}?$ $$ \text{a}.~\frac{3}{2}a_0 \qquad \text{b}.~\frac{3}{4}a_0 \qquad \text{c}.~3a_0 \qquad \text{d}.~\frac{1}{2}a_0 $$
I have written the normalized wavefunction of $\mathrm{1s}$ orbital of $\ce{He+}:$
$$R_{(1,0)} = \frac{2 \sqrt 2}{\sqrt{\pi a_0^3}} \times a^{-2r/a_0},$$
but I could not proceed further.
The average value of $r$ can be found out, in case of $\text s$ orbitals by multiplying the volume of each thin spherical shell by the probability density at that $r$ and adding them up.
This can simply be accomplished by using integration as a limit of sum. We know that the probability density $\mathrm dP/\mathrm dV = R^2$. So,
$$ \begin{align} ⟨r⟩ &= \int_0^\infty r\,\mathrm dP \\ ⟨r⟩ &= \int_0^\infty rR^2\,\mathrm {d}V\\ ⟨r⟩ &= \int_0^\infty r\left(\frac{2 \sqrt 2}{\sqrt{\pi a_0^3}} \times e^{-2r/a_0}\right)^2\left(4\pi r^2\,\mathrm {d}r\right)\\ ⟨r⟩ &= \frac{3a_0}{4}\\ \end{align} $$
