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Integrală Riemann

În analiza matematică, integrala Riemann constituie prima definiție riguroasă a integralei unei funcții pe un interval. A fost formulată de Bernhard Riemann și se poate aplica pentru funcții continue sau funcții regulate.

Preliminarii

Fie $[a,b]$ un interval (închis și mărginit), $a\leq b.$ O familie finită de puncte $d:(x_{0},x_{1},x_{2},\cdots ,x_{n}),$ astfel că:

a=x_{0}\leq x_{1}\leq x_{2}\leq \cdots \leq x_{i}\leq x_{i+1}\leq \cdots \leq x_{n-1}\leq x_{n}=b

se numește diviziune a intervalului $[a,b].$ Fiecare din intervalele $[x_{i},x_{i+1}]$ se numește interval parțial al diviziunii d.

Lungimea celui mai mare interval parțial al unei diviziuni $d:\;(x_{0},x_{1},\cdots ,x_{i},x_{i+1},\cdots ,x_{n})$ se numește norma diviziunii d și se notează:

\nu (d)=\max _{0\leq i\leq n-1}(x_{i+1}-x_{i}).

Definiție

Se spune că funcția f este integrabilă (în sensul lui Riemann) pe intervalul $[a,b]$ , dacă pentru orice șir de diviziuni $(d_{n})$ cu norma tinzând către zero și pentru orice alegere a punctelor intermediare $\xi _{i},$ șirurile corespunzătoare $(\sigma _{d_{n}})$ de sume integrale au o limită comună I.

Numărul I se numește integrala funcției f pe intervalul $[a,b]$ (în sensul lui Riemann) și se notează:

I=\int _{a}^{b}f(x)dx.

Notația $\int _{a}^{b}f(x)dx$ se citește "integrală de la a la b din f(x)dx".

Proprietăți

$1^{\circ }.\quad \int _{a}^{b}f(x)dx=-\int _{b}^{a}f(x)dx.$

$2^{\circ }.\quad \int _{a}^{a}f(x)dx=0.$

$3^{\circ }.\quad \int _{a}^{b}mdx=m(b-a).$

$4^{\circ }.\quad \int _{a}^{b}[mf(x)+ng(x)]dx=m\int _{a}^{b}f(x)dx+n\int _{a}^{b}g(x)dx$ oricare ar fi $m,n\in \mathbb {R} .$

$5^{\circ }.$ Dacă f și g sunt integrabile pe [a, b] și dacă $f(x)\leq g(x),\;\forall x\in [a,b]$ atunci $\int _{a}^{b}f(x)dx\leq \int _{a}^{b}g(x)dx.$

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