Definition 7.1.8: The Riemann Integral

Suppose f is a bounded function defined on a closed, bounded interval . Define the upper and lower Riemann integrals, respectively, as

I^*(f) = inf{ U(f,P): P a partition of [a, b]}
I_*(f) = sup{ L(f,P): P a partition of [a, b]}

Then if I^*(f) = I_*(f) the function f is called Riemann integrable and the Riemann integral of f over the interval [a, b] is denoted by

f(x) dx

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