Integral Cheat Sheet

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# Integral Cheat Sheet

## Definition

There are multiple definitions of integrals, but we’ll use the Darboux integral which is often mistakenly called Riemann integral.

Consider any partition $P$ of the line interval $[a, b]$ as:

$a = x_0 \lt x_1 \lt \dots \lt x_n = b$

Each $[x_{i-1}, x_{i}]$ for $i = 1, \dots, n$ is called a subinterval of $P$, which is also denoted by $(x_0, x_1, \dots, x_n)$.

Let $f: [a, b] \rightarrow \mathbb{R}$ and $P$ a partition of $[a, b]$. We define the upper $M_i$ and lower bounds $m_i$ of $f$ at a given subinterval as follows:

\begin{align} M_i &= \sup \curly{f(x) : x \in [x_{i-1}, x_{i}]} \\ m_i &= \inf \curly{f(x) : x \in [x_{i-1}, x_{i}]} \end{align}

The upper and lower Darboux sums are defined as:

\begin{align} U(f,P) &= \sum_{i = 1}^n M_i (x_{i-1} - x_{i}) \\ L(f,P) &= \sum_{i = 1}^n m_i (x_{i-1} - x_{i}) \end{align}

If we think of numerical analysis, $U(f,P)$ is an upper bound value for the area under the curve $f(x)$, by approximating it via rectangles. Similarly $L(f, P)$ is the lower bound.

Now let $\cal{P}$ the set of all possible partitions of $[a, b]$. The upper and lower Darboux integrals are defined as:

\begin{align} U(f) &= \inf \curly{U(f, P) : P \in \cal{P}} \\ L(f) &= \sup \curly{L(f, P) : P \in \cal{P}} \end{align}

Intuitively, $U(f)$ is the tighest upper bound for the area under the curve $f(x)$ and $L(f)$ is the tighest lower bound. If $U(f) = L(f)$, then we have an exact estimate for the area under the curve $f(x)$. The integral exists and $f$ is said to be Darboux integrable.