Integration described by first-order logic?

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Is it possible, to describe a Riemann integral with just first-order logic? And if so could someone point me to somewhere that has such a description of it.
 
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As far as I know any description of integration requires some qualifier of the form "##\forall A \subset X##" which is not expressible in first order logic. But I'd be happy to be proven wrong!
 
I don't know much formal logic beyond my intro model theory class, but...

In my attempt to define it, I couldn't do it without being able to say ``\exists n\in\mathbb N:\exists x_1,...\exists x_n:\enspace...\text{''} or ``\bigvee_{n=1}^\infty \exists x_1,...\exists x_n:\enspace...\text{''} which isn't kosher in any version of FOL I know.
 

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