Definition 7.1.5: Upper and Lower Sum

Let P = { x₀, x₁, x₂, ..., x_n} be a partition of the closed interval [a, b] and f a bounded function defined on that interval. Then:

The upper sum of f with respect to the partition P is defined as:

U(f, P) = c_j (x_j - x_j-1)

where c_j is the supremum of f(x) in the interval [x_j-1, x_j].

The lower sum of f with respect to the partition P is defined as

L(f, P) = d_j (x_j - x_j-1)

where d_j is the infimum of f(x) in the interval [x_j-1, x_j].

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