I am having some difficulty in understanding notation/meaning in the definition of definite integral. Can you guys help clarify/correct and fill in the gaps in my understanding.
Following is from wikipedia entry for the Darboux integral which is similar to what is in my textbook. My notes are inline.
A partition of an interval $[a,b]$ is a finite sequence of values $x_i$ such that,
$a = x_0 < x_1 < ... < x_n = b $
Partition to integrate over
Each interval $[x_i−1,x_i]$ is called a subinterval of the partition. Let $ƒ:[a,b]→R$ be a bounded function, and let
$P = (x_0, ..., x_n)$
be a partition of $[a,b]$. Let
Defining rectangles that would be used to approximate area under curve
$M_i = \sup\limits_{x \in [x_{i-1}, x_i]} f(x)$
$m_i = \inf\limits_{x \in [x_{i-1}, x_i]} f(x)$
Height of rectangle when above the curve or below the curve respectively
The upper Darboux sum of ƒ with respect to P is
$U_{f,P} = \sum_{i=1}^n (x_i - x_{i-1})M_i$
sum when using supremum for height
The lower Darboux sum of ƒ with respect to P is
$L_{f,P} = \sum_{i=1}^n (x_i - x_{i-1})m_i$
sum when using infimum for height
The upper Darboux integral of ƒ is
$U_f = \inf \{U_{f,P} : \mbox{P is a partition of} [a,b]\}$
The lower Darboux integral of ƒ is
$L_f = \sup \{L_{f,P} : \mbox{P is a partition of} [a,b]\}$
This is the part that I am unable to understand. What do the infimum and supremum imply here? I understand the definite integral as area under curve. Wouldn't the sums $U_{f,P}$ and $L_{f,P}$ then be the integrals themselves? Why do we need to take the infimum/supremum of these areas?
If $U_ƒ = L_ƒ$, then we say that $ƒ$ is Darboux-integrable and set
$\int_a^b f(t)dt = U_f = L_f$
this parts seems to intuitively suggest both areas would be nearly equal
Thanks for your help.