🍈 Zettelkasten

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Apr 19, 20252 min read

Notation

$\int_{a}^{b} f (x) d x$

$f (x)$ is referred to as the integrand
$d x$ is the differential The process to evaluate this integral is called integration

Given $a, b \in R, a < b$ , $f (x)$ and $g (x)$ are integrable on $[a, b]$

$\forall x \in [a, b], f (x) \geq 0 ⟹ \int_{b}^{a} f (x) d x \geq 0$
$\forall x \in [a, b], f (x) \leq 0 ⟹ \int_{b}^{a} f (x) d x \leq 0$
If $f + g$ is integrable on $[a, b]$ , then $\int_{a}^{b} [f (x) + g (x)] d x - \int_{a}^{b} f (x) d x = \int_{a}^{b} g (x) d x$
$\forall c \in R, \int_{a}^{b} c f (x) d x = c \int_{a}^{b} f (x) d x ⟹$ $c f (x)$ is integrable on $[a, b]$
$\int_{a}^{a} f (x) d x = 0$
$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
$\forall c \in (a, b), \int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x$ Union Interval Property