We have studied the integral and local accuracy of the WENO schemes in calculations with shocks propagating with a variable velocity. The integral accuracy is defined as the convergence order of spatial integrals of the difference solution calculated by the trapezoid or parabolic formulas. The local accuracy is associated with the determination of the errors in the calculation of invariants of exact solution. It is shown that when using the trapezoid formula, both the WENO3 and WENO5 schemes have only the first order of integral convergence in the intervals with one of their boundaries is in the shock influence region. If the parabolic formula is used, then the order of integral convergence for the WENO schemes decreases sharply in the intervals containing whole area of the shock wave influence. As a result, in these schemes, the local accuracy of the invariant calculations sharply decreases in the shock influence area. Moreover, in this area the local accuracy of the WENO3 and WENO5 schemes is approximately the same and much lower than the accuracy of the combined scheme, in which the Rusanov scheme is basic and the CABARET scheme is internal.