Though in general we'll be computing integrals in terms of their interpretation as areas, I gave the definition as a limit of Riemann sums. Since the limit in question is a limit over refinements of partitions (not the simple sort of limit of real numbers one uses to compute derivatives), I talked about limits in general for a while first.
Most substantively going forward, I stated a few of the nice properties integrals have, like the integral of a sum being the sum of the integrals, the integral of C*f being C * the integral of f (for C a constant), and that one can break the interval being integrated over into two pieces and add them up.
This motivates the definition we'll use for integrating "from a to b" when b < a, namely the negative of integrating from b to a.
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