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Given \(f:
\mathbb{R}\to\mathbb{R}\). Integral calculus: Area \(\int\limits_a^b f(x)\,\mathrm{d} x\) under
\(f\) in \([a,b]\).
\(a\), \(b\) limits (or bounds) of
integration
\(f\) integrand
\(x\) variable if
integration
\(\int_a^b f(x)\,\mathrm{d} x =
\int_a^b f(t)\,\mathrm{d}t = \int_a^b f(\heartsuit)\,\mathrm{d}
\heartsuit\)
Linearity
For \(\alpha,
\beta\in\mathbb{R}\): \(\int_a^b
(\alpha f+\beta g)(x)\,\mathrm{d} x = \alpha \int_a^b f(x)\,\mathrm{d} x
+ \beta \int_a^b g(x)\,\mathrm{d} x\)
Positivity and
monotonicity
\(f\geq 0\) \(\implies\) \(\int_a^b f(x)\,\mathrm{d} x\geq
0\)
\(f\leq g\) \(\implies\) \(\int_a^b f(x)\,\mathrm{d} x \leq \int_a^b
g(x)\,\mathrm{d} x\)
\(\int_a^b f(x)\,\mathrm{d} x =
\textcolor{green}{\text{area where} f\geq 0} -
\textcolor{blue}{\text{area where} f<0}\)
Partition and
absolute value
For \(a<c<b\): \(\int_a^b f(x)\,\mathrm{d} x =
\textcolor{green}{\int_a^c f(x)\,\mathrm{d} x} +
\textcolor{blue}{\int_c^b f(x)\,\mathrm{d} x}\)
\(\int_{b}^{a} f(x)\,\mathrm{d} x := -
\int_{a}^{b} f(x)\, \mathrm{d} x\), and \(\int_a^a f(x)\,\mathrm{d} x = 0\)
\(\left|\int_a^b f(x)\,\mathrm{d} x
\right| \leq \int_a^b \left|f(x)\right|\,\mathrm{d} x\)
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