{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "ac95962d-58e2-4c73-be9b-b33d20eb82ca",
"metadata": {},
"outputs": [],
"source": [
"# Questa cella carica tutte le librerie utili nelle seguenti parti di codice\n",
"# Markdown in code\n",
"from IPython.display import Markdown as md\n",
"from IPython.display import Latex\n",
"# Image mainipulation\n",
"from IPython.display import display # Modulo con strumenti di visualizzazione in IPython\n",
"# widgets\n",
"import ipywidgets as widgets\n",
"from ipywidgets import HBox, VBox\n",
"# numpy\n",
"import numpy as np\n",
"# matplotlib\n",
"import matplotlib.pyplot as plt\n",
"from matplotlib.patches import Rectangle\n",
"# %matplotlib inline"
]
},
{
"cell_type": "markdown",
"id": "59a8fed2-150b-4f5f-8dd6-d556add02902",
"metadata": {},
"source": [
"# Integrazione Numerica\n",
"\n",
"## Credits and Rights\n",
"\n",
"Questo notebook è stato originariamente creato dal Prof. Mirco Zerbetto; variazioni sono state apportate da Agostino Migliore.\n",
"\n",
"## 1 Introduzione\n",
"Il calcolo di aree (o volumi, o ipervolumi) è una problematica che si riscontra quando la quantità che si vuole determinare rappresenta una misura cumulativa di una osservabile del sistema che si sta analizzando in funzione di una o più variabili da cui dipende l'osservabile. Di seguito si riporta qualche esempio. \n",
"\n",
"##### Esempio 1. Determinazione della quantità di un analita da una misura HPLC\n",
"\n",
"Per determinare la quantità di un analita tramite HPLC si devono preparare delle soluzioni di quell'analita (possibilmente nella medesima matrice) che abbiano concentrazione nota e variabile. Dalla misura HPLC di ciascuno di tali standard si determina una relazione tra l'**area** del picco e la concentrazione dell'analita nello standard. Nota la relazione lineare, si esegue la misura sul campione e una volta determinata l'area del picco, si risale alla concentrazione dell'analita nel campione tramite la retta di taratura.\n",
"\n",
"##### Esempio 2. Determinazione della resa quantica di fluorescenza\n",
"\n",
"Per determinare la resa quantica di fluorescenza di una molecola bisogna preparare una soluzione di una sostanza per la quale la resa quantica sia nota nelle condizioni sperimentali (matrice). La resa quantica è proporzionale al rapporto tra le aree dei picchi di emissione della sostanza nota e di quella incognita.\n",
"\n",
"##### Esempio 3. Determinazione della costante di *binding* di un substrato a un enzima\n",
"\n",
"Per determinare la costante di equilibrio della formazione di un addotto enzima-substrato si usa la titolazione calorimetrica isoterma (ITC, *isothermal titration calorimetry*) in cui si misura il calore sviluppato (o assorbito) dalla reazione di *binding* via via che si titola l'enzima con il substrato. Il grafico che mette in relazione il tasso di variazione del calore in funzione del tempo dopo ogni aggiunta è costituito da molti picchi. L'area sottesa a ciascuno dei picchi è il calore sviluppato nel lasso di tempo in cui il sistema, a seguito dell'aggiunta del substrato, ha raggiunto un nuovo stato di equilibrio."
]
},
{
"cell_type": "markdown",
"id": "7c4f119e-e7fd-4083-b0e3-86034032e002",
"metadata": {},
"source": [
"## 2 Calcolo di integrali per via numerica\n",
"In tutti gli esempi sopra citati si deve calcolare l'integrale\n",
"$$\n",
"I=\\int_a^b f(x)dx\n",
"$$\n",
"dove $\\Omega=(b-a)$ è il dominio di integrazione, mentre $f(x)$ in tutta generalità è una funzione della quale non si conosce l'espressione analitica o per la quale è difficile (o impossibile) calcolare la primitiva. Lo scopo è trovare un'approssimazione numerica all'integrale."
]
},
{
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"
}
},
"cell_type": "markdown",
"id": "206dd1a0-fa25-4cba-a984-9a5917bc1c49",
"metadata": {},
"source": [
" Geometricamente, l'integrale sopra riportato è l'area sottesa alla funzione $f(x)$ entro il dominio $\\Omega$. \n",
"
\n",
"\n",
"