Integral definida

Actualmente

Definition (Darboux sums)

Let $f: [a, b] \to \mathbb{R}$ be a bounded function. Let $P$ be a partition of $[a, b]$, $I_n = [x_{i-1}, x_{i}]$, $\Delta x_i = x_{i} - x_{i-1}$, then

• Upper Darboux sum: $\displaystyle U(f,P) = \sum_{i=1}^{n} \left(\sup_{x \in I_i} \{f(x)\} \cdot \Delta x_i \right)$


• Lower Darboux sum: $\displaystyle L(f,P) = \sum_{i=1}^{n} \left(\inf_{x \in I_i} \{f(x)\} \cdot \Delta x_i \right)$

Definition (Darboux integrals)

Let $f: [a, b] \to \mathbb{R}$ be a bounded function. Let

• Upper Darboux integral: $U(f) = \inf \{U(f,P) : \text{ for all partition } P \}$


• Lower Darboux integral: $L(f) = \sup \{L(f,P) : \text{ for all partition } P \}$

Definition (Riemann sum)

Let $f: [a, b] \to \mathbb{R}$ be a bounded function. Let $P$ be a partition of $[a, b]$ and $X$ a selection of points in each subinterval $c_i \in [x_{i-1}, x_{i}]$. Let $\Delta x_i = x_{i} - x_{i-1}$. The associated Riemann sum is $$S(f, P, X) = \sum_{i = 1}^{n} f(c_i) \Delta x_i$$

Definition (Riemann integral)

Let $f: [a, b] \to \mathbb{R}$ be a bounded function. The function $f$ is Riemann integrable on $[a, b]$ if and only if $$(\forall \varepsilon > 0)(\exists \delta > 0)(\forall P)(\forall X): \|P\|\ \leq \delta \implies | I - S(f, P, X) | \leq \varepsilon$$

Theorem (Riemann's criterion)

Let $f: [a, b] \to \mathbb{R}$ be a bounded function. The function $f$ is Riemann integrable on $[a, b]$ if and only if $$(\forall \varepsilon > 0)(\exists P): U(f, P) - L(f, P) \leq \varepsilon$$

Theorem (Darboux's criterion)

Let $f: [a, b] \to \mathbb{R}$ be a bounded function. The function $f$ is Riemann integrable on $[a, b]$ if and only if $$(\forall \varepsilon > 0)(\exists \delta > 0)(\forall P): \|P\|\ \leq \delta \implies U(f, P) - L(f, P) \leq \varepsilon$$

Theorem

Let $f: [a, b] \to \mathbb{R}$ be a bounded function, $P_1, P_2$ be partitions of $[a, b]$, then $$L(f, P_1) \leq U(f, P_2)$$

Theorem

Let $f: [a, b] \to \mathbb{R}$ be a bounded function, $P_1, P_2$ be partitions of $[a, b]$ where $P_2$ is a refinement of $P_1$, then $$L(f, P_1) \leq L(f, P_2) \leq U(f, P_2) \leq U(f, P_1)$$

Theorem

Let $f: [a, b] \to \mathbb{R}$ be a bounded function, then $L(f) \leq U(f).$