Integral

analysis

Definite Integral

Definition

Definitive Integral

Let $I=[a,b]$ an interval and $f:I\to \mathbb{R}$. If every sequence $(S_n{)}_{n\in \mathbb{N}}$ of Riemann sums, whose corresponding sequence of partitions satisfies ${\lim }_{n\to \infty }\mathcal{F}(Z_n)=0$, converges to the same limit, then this limit is called the definite integral of $f$ on the interval $[a,b]$ and is written as ${\int }_a^{b}f(x)dx$.

Here, $a$ and $b$ are called the limits of integration (or bounds of integration) and $x$ is called the integration variable. If the upper limit of integration is not greater than the lower limit, one defines:

$$\begin{array}{rl}{\int }_a^{b}f(x)dx& =-{\int }_b^{a}f(x)dx\text{if }a>b\\ {\int }_a^{a}f(x)dx& =0\end{array}$$

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Indefinite Integral

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