Answer by Roland Bacher for What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Replacing the characteristic function of the unit ball by a suitable normal distribution with spherical symmetry when computing the volume should give approximatively the correct answer.Since $$\frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n}(x_1^2+\dots+x_n^2)e^{-(x_1^2+\dots+x_n^2)/2}dx_1\cdots dx_n$$is linear in $n$, one has to rescale by a factor of order $\frac{1}{\sqrt{n}}$leading to a decay of order $(\lambda n)^{-n/2}$ for the volume of the unit sphere.