  We prove the curse of dimensionality in the worst case setting for numerical
integration for a number of classes of smooth $d$-variate functions. Roughly
speaking, we consider different bounds for the derivatives of $f \in C^k(D_d)$
and ask whether the curse of dimensionality holds for the respective classes of
functions. We always assume that $D_d \subset \mathbb{R}^d$ has volume one and
consider various values of $k$ including the case $k=\infty$ which corresponds
to infinitely many differentiable functions. We obtain necessary and sufficient
conditions, and in some cases a full characterization for the curse of
dimensionality. For infinitely many differentiable functions we prove the curse
if the bounds on the successive derivatives are appropriately large. The proof
technique is based on a volume estimate of a neighborhood of the convex hull of
$n$ points which decays exponentially fast if $n$ is small relative to $d$. For
$k=\infty$, we also also study conditions for quasi-polynomial, weak and
uniform weak tractability.
