  We study lower bounds on the worst-case error of numerical integration in
tensor product spaces. As reference we use the $N$-th minimal error of linear
rules that use $N$ function values. The information complexity is the minimal
number $N$ of function evaluations that is necessary such that the $N$-th
minimal error is less than a factor $\varepsilon$ times the initial error. We
are interested to which extent the information complexity depends on the number
$d$ of variables of the integrands. If the information complexity grows
exponentially fast in $d$, then the integration problem is said to suffer from
the curse of dimensionality.
  Under the assumption of the existence of a worst-case function for the
uni-variate problem we present two methods for providing good lower bounds on
the information complexity. The first method is based on a suitable
decomposition of the worst-case function. This method can be seen as a
generalization of the method of decomposable reproducing kernels, that is often
successfully applied when integration in Hilbert spaces with a reproducing
kernel is studied. The second method, although only applicable for positive
quadrature rules, has the advantage, that it does not require a suitable
decomposition of the worst-case function. Rather, it is based on a spline
approximation of the worst-case function and can be used for analytic
functions.
  The methods presented can be applied to problems beyond the Hilbert space
setting. For demonstration purposes we apply them to several examples, notably
to uniform integration over the unit-cube, weighted integration over the whole
space, and integration of infinitely smooth functions over the cube. Some of
these results have interesting consequences in discrepancy theory.
