数学 - Python - JavaScript - 解析学 - 積分 - 積分法 - 上方和および下方和(累乗(べき乗、2乘)、関数、区間、分割)

解析入門 原書第3版
原著

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第3部(積分)、第9章(積分法)、5(上方和および下方和)、練習問題1.を取り組んでみる。

1. 1. 
      

      上方和。

      $\begin{array}{}\frac{1}{2}\left({\left(\frac{3}{2}\right)}^2+2^2\right)\\ =\frac{1}{2}\left(\frac{9}{4}+4\right)\\ =\frac{25}{8}\end{array}$

      下方和。

      $\begin{array}{}\frac{1}{2}\left(1^2+{\left(\frac{3}{2}\right)}^2\right)\\ =\frac{1}{2}\left(1+\frac{9}{4}\right)\\ =\frac{13}{8}\end{array}$
   2. 
      

      上方和。

      $\begin{array}{}\frac{1}{3}\left({\left(\frac{4}{3}\right)}^2+{\left(\frac{5}{3}\right)}^2+2^2\right)\\ =\frac{1}{3}·\; <!--\; middle\; dot\; -->\frac{16+25+36}{9}\\ =\frac{1}{3}\frac{77}{9}\\ =\frac{77}{27}\end{array}$

      下方和。

      $\begin{array}{}\frac{1}{3}\left({\left(\frac{3}{3}\right)}^2+{\left(\frac{4}{3}\right)}^2+{\left(\frac{5}{3}\right)}^2\right)\\ =\frac{1}{3}·\; <!--\; middle\; dot\; -->\frac{9+16+25}{9}\\ =\frac{50}{27}\end{array}$
   3. 
      

      上方和。

      $\begin{array}{}\frac{1}{4}\left({\left(\frac{5}{4}\right)}^2+{\left(\frac{6}{4}\right)}^2+{\left(\frac{7}{4}\right)}^2+{\left(\frac{8}{4}\right)}^2\right)\\ =\frac{1}{4}·\; <!--\; middle\; dot\; -->\frac{1}{16}\frac{1}{6}·\; <!--\; middle\; dot\; -->\left(8\left(8+1\right)\left(16+1\right)-4\left(4+1\right)\left(8+1\right)\right)\\ =\frac{1}{16·\; <!--\; middle\; dot\; -->6}\left(2·\; <!--\; middle\; dot\; -->9·\; <!--\; middle\; dot\; -->17-5·\; <!--\; middle\; dot\; -->9\right)\\ =\frac{1}{16}·\; <!--\; middle\; dot\; -->\frac{1}{2}\left(2·\; <!--\; middle\; dot\; -->3·\; <!--\; middle\; dot\; -->17-15\right)\\ =\frac{102-15}{32}\\ =\frac{87}{32}\end{array}$

      下方和。

      $\begin{array}{}\frac{1}{4}\left({\left(\frac{4}{4}\right)}^2+{\left(\frac{5}{4}\right)}^2+{\left(\frac{6}{4}\right)}^2+{\left(\frac{7}{4}\right)}^2\right)\\ =\frac{1}{4}·\; <!--\; middle\; dot\; -->\frac{1}{16}·\; <!--\; middle\; dot\; -->\frac{1}{6}\left(7·\; <!--\; middle\; dot\; -->8·\; <!--\; middle\; dot\; -->15-3·\; <!--\; middle\; dot\; -->4·\; <!--\; middle\; dot\; -->7\right)\\ =\frac{1}{16}·\; <!--\; middle\; dot\; -->\frac{1}{6}\left(7·\; <!--\; middle\; dot\; -->2·\; <!--\; middle\; dot\; -->15-3·\; <!--\; middle\; dot\; -->7\right)\\ =\frac{1}{16}·\; <!--\; middle\; dot\; -->\frac{7}{2}\left(2·\; <!--\; middle\; dot\; -->5-7\right)\\ =\frac{7}{32}·\; <!--\; middle\; dot\; -->3\\ =\frac{63}{32}\end{array}$
   4. 
      

      上方和。

      $\begin{array}{}\frac{1}{n}\left({\left(1+\frac{1}{n}\right)}^2+{\left(1+\frac{2}{n}\right)}^2+\dots \; <!--\; horizontal\; ellipsis\; -->+{\left(1+\frac{n}{n}\right)}^2\right)\\ =\frac{1}{n}\left({\left(\frac{n+1}{n}\right)}^2+{\left(\frac{n+2}{n}\right)}^2+\dots \; <!--\; horizontal\; ellipsis\; -->+{\left(\frac{n+n}{n}\right)}^2\right)\\ =\frac{1}{n^3}·\; <!--\; middle\; dot\; -->\frac{1}{6}\left(\left(2n\right)\left(2n+1\right)\left(4n+1\right)-n\left(n+1\right)\left(2n+1\right)\right)\\ =\frac{1}{6n^2}·\; <!--\; middle\; dot\; -->\left(2n+1\right)\left(2\left(4n+1\right)-\left(n+1\right)\right)\\ =\frac{\left(2n+1\right)\left(7n+1\right)}{6n^2}\end{array}$

      下方和。

      $\begin{array}{}\frac{1}{n}\left(1^2+{\left(1+\frac{1}{n}\right)}^2+\dots \; <!--\; horizontal\; ellipsis\; -->+{\left(1+\frac{n-1}{n}\right)}^2\right)2\\ =\frac{1}{n}\left(\frac{n^2}{n^2}+\frac{{\left(n+1\right)}^2}{n^2}+\dots \; <!--\; horizontal\; ellipsis\; -->+\frac{\left(2n-1\right)}{n^2}\right)\\ =\frac{1}{n^3}\frac{1}{6}\left(\left(2n-1\right)\left(2n\right)\left(4n-2+1\right)-\left(n-1\right)n\left(2n-2+1\right)\right)\\ =\frac{1}{6n^3}\left(\left(2n-1\right)\left(2n\right)\left(4n-1\right)-\left(n-1\right)n\left(2n-1\right)\right)\\ =\frac{1}{6n^2}\left(2n-1\right)\left(8n-2-n+1\right)\\ =\frac{\left(2n-1\right)\left(7n-1\right)}{6n^2}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, summation

i, n = symbols('i, n', integer=True)
x = symbols('x')
f = x ** 2
u = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 1, n))
l = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 0, n - 1))

for n0 in [2, 3, 4]:
    d = {n: n0}
    for g in [u, l]:
        pprint(g.subs(d))
        print()
    print()

for t in [u, l]:
    pprint(t.factor())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
25/8

13/8


77
──
27

50
──
27


87
──
32

63
──
32


(2⋅n + 1)⋅(7⋅n + 1)
───────────────────
           2       
        6⋅n        

(2⋅n - 1)⋅(7⋅n - 1)
───────────────────
           2       
        6⋅n        

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="2">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample1.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_n0],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let f = (x) => x ** 2;

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        n0 = parseInt(input_n0.value, 10);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [],
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [];

    for (let i = 1; i <= 2; i += 1 / n0) {
        lines.push([i, y1, i, y2, 'red']);
    }
    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
    
    fns2
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
n =