Editing Darboux integral (section)

===Darboux sums===
[[Image:Darboux.svg|thumb|right|Lower (green) and upper (green plus lavender) Darboux sums for four subintervals|350x350px]]
A [[partition of an interval]] <math>[a,b]</math> is a finite sequence of values <math>x_{i}</math> such that

:<math>a = x_0 < x_1 < \cdots < x_n = b.</math>

Each interval <math>[x_{i-1},x_i]</math> is called a ''subinterval'' of the partition. Let <math>f:[a,b]\to\R</math> be a bounded function, and let

:<math>P = (x_0, \ldots, x_n)</math>

be a partition of <math>[a,b]</math>. Let

:<math>\begin{align}
M_i = \sup_{x\in[x_{i-1},x_{i}]} f(x), \\
m_i = \inf_{x\in[x_{i-1},x_{i}]} f(x).
\end{align}</math>


The '''upper Darboux sum''' of <math>f</math> with respect to <math>P</math> is

:<math>U_{f, P} = \sum_{i=1}^n (x_{i}-x_{i-1}) M_i. \,\!</math>

The '''lower Darboux sum''' of <math>f</math> with respect to <math>P</math> is

:<math>L_{f, P} = \sum_{i=1}^n (x_{i}-x_{i-1}) m_i. \,\!</math>

The lower and upper Darboux sums are often called the lower and upper sums.