On the one hand, one has the various functional calculi from Operator Algebras. The continuous functional calculus for C* algebras, the bounded borel functional calculus for Von Neumann Algebras, the analytic one for Banach algebras.
On the other hand, one has the notions of integral and derivative (analyticity too, etc.) as the usual definitions for Banach-space-valued functions. (There are the Bochner integrals, as well as the Riemann integral for the continuous case.)
Are these two things related? I often see the functional calculus denoted with integrals, so I don't know if that's suggestive.