  Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain
integrals over $T$, one of which was computed by Euler. We give expressions for
them both as a linear combination of multiple zeta values, and as a polynomial
in single zeta values. We obtain asymptotic expansions of the integrals, and of
sums of certain multiple zeta values with constant weight. We also give related
expressions for Euler's constant. In the final section, we evaluate more
general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral --
over some polytopes that are higher-dimensional analogs of $T$. This leads to a
relation between certain multiple polylogarithm values and multiple zeta
values.
