Integration of monomials over the unit spere and unit ball in $R^n$

We compute the integral of monomials of the form $x^{2\beta }$ over the unit sphere and the unit ball in $R^n$ where $\beta =({\beta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\beta }_n)$ is a multi-index with real components ${\beta }_k>-1\mathrm{/}2$, $1\le k\le n$, and discuss their asymptotic behavior as some, or all, ${\beta }_k\to \mathrm{\infty }$. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in $R^n$. We also consider the Fourier transform of monomials $x^{\alpha }$ restricted to the unit sphere in $R^n$, where the multi-indices $\alpha$ have integer components, and discuss their behaviour at the origin.