  We present a method to derive asymptotics of eigenvalues for trace-class
integral operators $K:L^2(J;d\lambda)\circlearrowleft$, acting on a single
interval $J\subset\mathbb{R}$, which belong to the ring of integrable operators
\cite{IIKS}. Our emphasis lies on the behavior of the spectrum
$\{\lambda_i(J)\}_{i=0}^{\infty}$ of $K$ as $|J|\rightarrow\infty$ and $i$ is
fixed. We show that this behavior is intimately linked to the analysis of the
Fredholm determinant $\det(I-\gamma K)|_{L^2(J)}$ as $|J|\rightarrow\infty$ and
$\gamma\uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic
formul\ae\, are obtained for the eigenvalues of Airy and Bessel kernels in
random matrix theory.
