We compute the integral of monomials of the form x^2β over the unit sphere and the unit ball in R^n where β = (β1, . . . , βn) is a multi–index with real components βk > −1/2, 1 ≤ k ≤ n, and discuss their asymptotic behavior as some, or all, βk → ∞. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in Rn. We also consider the Fourier transform of monomials xα restricted to the unit sphere in Rn, where the multi–indices α have integer components, and discuss their behaviour at the origin.