  We introduce an adaptation of integral approximation operators to set-valued
functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the
space of compact non-empty subsets of ${\mathbb R}^d$. All operators are
adapted by replacing the Riemann integral for real-valued functions by the
weighted metric integral for SVFs of bounded variation with compact graphs. For
such a set-valued function $F$, we obtain pointwise error estimates for
sequences of integral operators at points of continuity, leading to convergence
at such points to $F$. At points of discontinuity of $F$, we derive estimates,
which yield the convergence to a set, first described in our previous work on
the metric Fourier operator. Our analysis uses recently defined one-sided local
quasi-moduli at points of discontinuity and several notions of local Lipschitz
property at points of continuity.
  We also provide a global approach for error bounds. A multifunction $F$ is
represented by the set of all its metric selections, while its approximation
(its image under the operator) is represented by the set of images of these
metric selections under the operator. A bound on the Hausdorff distance between
these two sets of single-valued functions in $L^1$ provides our global
estimates.
  The theory is illustrated by presenting the examples of two concrete
operators: the Bernstein-Durrmeyer operator and the Kantorovich operator.
