Infi-2 3-4

Definite integrals

Riemann sum #definition

\begin{matrix} Let [a, b] \subseteq R \\ Let P = {x_{0}, x_{1}, \dots, x_{n}} \\ a = x_{0} < x_{1} < \dots < x_{n} = b \\ Let C = {c_{i} | \forall i \in [1, n] : c_{i} \in [x_{i - 1} + x_{i}]} \\ Let Δ x_{i} = x_{i - 1} - x_{i} \\ S_{R} (f, P, C) = \sum_{i = 1}^{n} f (c_{i}) \cdot (x_{i} - x_{i - 1}) = \sum_{i = 1}^{n} f (c_{i}) Δ x_{i} \\ S_{R} (f, P, C) is called a Riemann sum and is an approximation of area under function f \\ This sum is the best approximation when all chosen rectangles’ areas tend to 0 \\ Let λ (P) = max {Δ x_{i}} \\ When λ (P) tends to 0, all Δ x tend to 0 and thus all rectangles’ areas tend to 0 \end{matrix}

Riemann-Integrable function #definition

\begin{matrix} Let f be a function on [a, b] \\ f is called integrable by Riemann if: \\ 1. f is bounded \\ 2. \forall P, C : S_{R} (f, P, C) \underset{λ (P) \to 0}{\to} L \\ Same definition via sequences: \\ f is integrable on [a, b] if \\ \forall {P_{n}} : λ (P_{n}) \underset{n \to \infty}{\underset{⏟}{\to}} 0 : \forall {C_{n}} : S_{R} (f, P_{n}, C_{n}) \to L \end{matrix}

Definite integral #definition

\begin{matrix} If such limit L exists, it is called definite integral of f on [a, b] \\ And denoted as \int_{a}^{b} f (x) d x = L \end{matrix}

Example

\begin{matrix} Let f (x) = C on [a, b] \\ S_{R} (f, P, C) = \sum_{i = 1}^{n} C Δ x_{i} = C \sum_{i = 1}^{n} Δ x_{i} \\ \forall P, C : \sum_{i = 1}^{n} Δ x_{i} = \sum_{i = 1}^{n} [x_{i} - x_{i - 1}] = x_{n} - x_{0} = b - a \\ ⟹ \int_{a}^{b} f (x) d x = C (b - a) \end{matrix}

Oscillation of a function #definition

\begin{matrix} Let f be a bounded function on I (for example on [a, b]) \\ Oscillation of f is a measure of how function varies between its extreme values \\ And is denoted as ω (I) = sup f - inf f \end{matrix}

Lebesgue criterion of Riemann-Integrability #theorem

\begin{matrix} Let f be a bounded function on [a, b] \\ Then f is integrable on [a, b] iff \\ \forall {P_{n}} : λ (P_{n}) \underset{n \to \infty}{\to} 0 : \sum_{i = 1}^{n} ω_{i} Δ x_{i} \to 0 \\ Where \forall i \in [1, n] : w_{i} = ω ([x_{i}, x_{i - 1}]) \\ Exaplanation (not proof): \\ ⟹ Let f be integrable on [a, b] \\ ⟹ All Riemann sums tend to L \\ \sum_{i = 1}^{n} ω_{i} Δ x_{i} = \sum_{i = 1}^{n} (sup_{[x_{i - 1}, x_{i}]} f - inf_{[x_{i - 1}, x_{i}]} f) Δ x_{i} = \\ = \underset{Upper Riemann sum}{\underset{⏟}{\sum_{i = 1}^{n} sup_{[x_{i - 1}, x_{i}]} f Δ x_{i}}} - \underset{Lower Riemann sum}{\underset{⏟}{\sum_{i = 1}^{n} inf_{[x_{i - 1}, x_{i}]} f Δ x_{i}}} \to L - L = 0 \\ Why is it not formal proof: \\ There might be no point c_{i} such that f (c_{i}) = sup_{[x_{i - 1}, x_{i}]} f \\ ⟹ These sums might not be Riemann sums at all \\ ⟸ Let \forall {P_{n}} : λ (P_{n}) \underset{n \to \infty}{\to} 0 : \sum_{i = 1}^{n} ω_{i} Δ x_{i} \to 0 \\ \sum_{i = 1}^{n} ω_{i} Δ x_{i} = \sum_{i = 1}^{n} (sup_{[x_{i - 1}, x_{i}]} f - inf_{[x_{i - 1}, x_{i}]} f) Δ x_{i} = \sum_{i = 1}^{n} sup_{[x_{i - 1}, x_{i}]} f Δ x_{i} - \sum_{i = 1}^{n} inf_{[x_{i - 1}, x_{i}]} f Δ x_{i} \\ \sum_{i = 1}^{n} ω_{i} Δ x_{i} \to 0 ⟹ {\begin{matrix} \sum_{i = 1}^{n} sup_{[x_{i - 1}, x_{i}]} f Δ x_{i} \to L \\ \sum_{i = 1}^{n} inf_{[x_{i - 1}, x_{i}]} f Δ x_{i} \to L \end{matrix} \\ c_{i} \in [x_{i - 1}, x_{i}] ⟹ inf_{[x_{i - 1}, x_{i}]} f \leq f (c_{i}) \leq sup_{[x_{i - 1}, x_{i}]} f \\ ⟹ \underset{\to L}{\underset{⏟}{\sum_{i = 1}^{n} inf_{[x_{i - 1}, x_{i}]} f Δ x_{i}}} \leq S_{R} (f, P, C) \leq \underset{\to L}{\underset{⏟}{\sum_{i = 1}^{n} sup_{[x_{i - 1}, x_{i}]} f Δ x_{i}}} \\ ⟹ S_{R} (f, P, C) \to L ⟹ f is Riemann-integrable \\ Why is it not formal proof: \\ x_{n} - y_{n} \to 0 ⟹ ⧸ {\begin{matrix} x_{n} \to L \\ y_{n} \to L \end{matrix} \\ x_{n} and y_{n} might not have a limit at all \end{matrix}