Definition 7.1.3: Riemann Sums

If P = { x₀, x₁, x₂, ..., x_n} is a partition of the closed interval [a, b] and f is a function defined on that interval, then the n-th Riemann Sum of f with respect to the partition P is defined as:

R(f, P) = f(t_j) (x_j - x_j-1)

where t_j is an arbitrary number in the interval [x_j-1, x_j].

Note: If t_i is always the left endpoint of each subinterval, the corresponding Riemann sum is called left Riemann sum; if it is always the right endpoint, the sum is called right Riemann sum.

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