We study cubature formulas for d-dimensional integrals with a high trigonom
etric degree. To obtain a trigonometric degree l in dimension d, we need ab
out d(l)/l! function values if d is large. Only a small number of arithmeti
cal operations is needed to construct the cubature formulas using Smolyak's
technique. We also compare different methods to obtain formulas with high
trigonometric degree.