Let {X-n,n greater than or equal to 1} be a sequence of iid. Gaussian random vectors in R-d, d greater than or equal to 2, with nonsingular distribution function F. In this paper the asymptotics for the sequence of integrals I-F,I-n(G(n)) := n integral(Rd)G(n)(n-1)(X) dF(X) is considered with G(n) some distribution function on R-d. In the case G(n) = F the integral I-F,I-n(F)/n is the probability that a record occurs in X-1,..., X-n at index n. (1) obtained lower and upper asymptotic bounds for this case, whereas (2) showed the rate of convergence if d = 2. In this paper we derive the exact rate of convergence of I-F,I-n(G(n)) for d greater than or equal to 2 under some restrictions on the distribution function Gn. Some related results for multivariate Gaussian tails are discussed also.