This paper studies the numerical computation of integrals, representing
estimates or predictions, over the output $f(x)$ of a computational model with
respect to a distribution $p(\mathrm{d}x)$ over uncertain inputs $x$ to the
model. For the functional cardiac models that motivate this work, neither $f$
nor $p$ possess a closed-form expression and evaluation of either requires
$\approx$ 100 CPU hours, precluding standard numerical integration methods. Our
proposal is to treat integration as an estimation problem, with a joint model
for both the a priori unknown function $f$ and the a priori unknown
distribution $p$. The result is a posterior distribution over the integral that
explicitly accounts for dual sources of numerical approximation error due to a
severely limited computational budget. This construction is applied to account,
in a statistically principled manner, for the impact of numerical errors that
(at present) are confounding factors in functional cardiac model assessment.