For a function \(f\) defined on \((a,b),\) the definite integral of \(f\) from \(a\) to \(b,\) denoted \( \int_a^b f(x) \,\mathrm{d}x\,, \) represents the signed area of the region bound by the graph of \(f\) and the \(x\)-axis between \({x=a}\) and \({x=b.}\) We refer to \(f\) as the integrand and to \(a\) and \(b\) as the lower bound and upper bound of the integral respectively. The definite integral counts the amount of area that \(f\) accumulates as \(x\) sweeps from \(a\) to \(b.\) The area is signed in the sense that any portion of the region underneath the \(x\)-axis counts negatively towards the accumulated area.
Formally, the definite integral is defined as the limit of a Riemann sum
as the number of summands approaches infinity,
and we say \(f\) is integrable only if this limit exists:
\[
\int\limits_a^b f(x)\,\mathrm{d}x
\;\;=\;\; \lim\limits_{n \to \infty} \sum_{i=1}^{n} \bigl(f(a+ih) \times h\bigr)
\quad \text{for}\;\; h = \frac{b-a}{n}
\]
The definite integral of \(f\) could also be computed
as the product \((f_{\text{avg}})\!\times\!(b\!-\!a)\)
where \(f_\text{avg}\) is the average value of \(f\) on the interval \([a,b].\)
Often though, its not possible to determine the average value of \(f\) directly,
so we use this fact as a means for computing the average value in terms of the definite integral:
\[
(f_{\text{avg}}) \times (b\!-\!a) = \int\limits_a^b f(x)\,\mathrm{d}x
\qquad\implies\qquad
f_{\text{avg}} = \frac{1}{b\!-\!a}\int\limits_a^b f(x)\,\mathrm{d}x
\]