Let T_d:L₂([0, 1]^d)→C([0, 1]^d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k^-1(logk)^d-1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]^d under the C([0, 1]^d)-norm can be estimated from below by exp(-Cε^-2logε^2d-1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.