Complex Riemann Integral is Contour Integral

Theorem
Let $f: \R \to \C$ be a complex Riemann integrable function over some closed real interval $\closedint a b$.

Then:
 * $\ds \int_a^b \map f t \rd t = \int_\CC \map f t \rd t$

where:
 * the integral on the is a complex Riemann integral
 * the integral on the is a contour integral
 * $\CC$ is a straight line segment along the real axis, connecting $a$ to $b$.