06. Definite Integrals

$\gdef \N{\mathbb{N}} \gdef \Z{\mathbb{Z}} \gdef \Q{\mathbb{Q}} \gdef \R{\mathbb{R}} \gdef \C{\mathbb{C}} \gdef \setcomp#1{\overline{#1}} \gdef \sseq{\subseteq} \gdef \pset#1{\mathcal{P}(#1)} \gdef \covariant{\operatorname{Cov}} \gdef \of{\circ} \gdef \p{^{\prime}} \gdef \pp{^{\prime\prime}} \gdef \ppp{^{\prime\prime\prime}} \gdef \pn#1{^{\prime\times{#1}}} $

Leading question:
How to calculate area under graph \(y=f(x)\) between \([a,b]\)?
Idea: Subdivide interval into \(n\) equal parts.
The length of each subinterval \(=\Delta x=\frac{b-a}n\).
\(x_k=a+k\Delta x\).
Now, build a rectangle on top of each subinterval \([x_k, x_{k+1}]\) where the height is \(f(x_{k+1})\)—i.e., a rectangle that touches the graph of \(f\) in the upper right corner.
As an approximation of the area under the graph, we take the sum of the areas of these rectangles:
\(S_n=f(x_1)\cdot\Delta x+f(x_2)\cdot\Delta x+\dots+f(x_n)\cdot\Delta x=\sum_{k=1}^n{f(x_k)\cdot\Delta x}\)
This expression for \(S_n\) is called a Reimann sum.
Now, let \(n\to\infty\); the more rectangles we have, the more precisely the area will fill exactly the area under the graph.
This limit of Reimann sums is called the definite integral of \(f\) from \(a\) to \(b\).