We compute the “surface areas” of n-dimensional spheres. Of course, we know that \int_{-\infty}^\infty e^{-t^2}\; dt=\sqrt{\pi}. Raising this to the d^\text{th} power gives \int_{\mathbb{R}^d} e^{-|x|^2}\; dx=\pi^{d/2}. Making the substitution dx=r^{d-1}\; dr\; d\omega gives \int_{S^{d-1}} d\omega\int_0^\infty e^{-r^2}r^{d-1}\; dr=|S^{d-1}|\int_0^\infty e^{-u}u^{d/2-1}\;\frac{du}{2}=\frac{|S^{d-1}|}{2}\Gamma\left(\frac{d}{2}\right). Hence |S^{d-1}|=\frac{2\pi^{d/2}}{\Gamma(d/2)}.

Taking the limit as d\to\infty yields a surprising consequence: the surface areas of spheres of radius 1 tend to 0 as the dimension increases!