数学 - Python - JavaScript - 細分による加法 - 積分法 – 定積分の性質と計算 - 簡単な例(絶対値、累乗(べき乗)、三角関数(正弦、2倍角)、指数関数)

数学読本〈5〉

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)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 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

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第19章(細分による加法 - 積分法)、19.3(定積分の性質と計算)、簡単な例、問22.を取り組んでみる。

22. 1. 
       $\begin{array}{l}{x}^2-1=0\\ x=\pm 1\\ {\displaystyle {\int }_0^1\left(1-x^2\right)dx+{\displaystyle {\int }_1^3\left(x^2-1\right)dx}}\\ ={\left[x-\frac{1}{3}{x}^3\right]}_0^1+{\left[\frac{1}{3}{x}^3-x\right]}_1^3\\ =1-\frac{1}{3}+9-3-\left(\frac{1}{3}-1\right)\\ =8-\frac{2}{3}\\ =\frac{22}{3}\end{array}$
    2. 
       $\begin{array}{l}{\displaystyle {\int }_{-1}^1\left|x\right|dx}-{\displaystyle {\int }_{-1}^1\left|x-1\right|dx}\\ =2{\displaystyle {\int }_0^{1}xdx}+{\displaystyle {\int }_{-1}^1\left(x-1\right)dx}\\ =2{\left[\frac{1}{2}{x}^2\right]}_0^1+{\displaystyle {\int }_{-1}^{1}xdx-{\displaystyle {\int }_{-1}^{1}1dx}}\\ =2·\frac{1}{2}+0-2\\ =-1\end{array}$
    3. 
       $\begin{array}{l}x-x^2\\ =-x\left(x-1\right)\\ \\ -{\displaystyle {\int }_{-1}^0\left(x-x^2\right)dx}+{\displaystyle {\int }_0^1\left(x-x^2\right)dx}-{\displaystyle {\int }_1^2\left(x-x^2\right)dx}\\ =-{\left[\frac{1}{2}{x}^2-\frac{1}{3}{x}^3\right]}_{-1}^0+{\left[\frac{1}{2}{x}^2-\frac{1}{3}{x}^3\right]}_0^1-{\left[\frac{1}{2}{x}^2-\frac{1}{3}{x}^3\right]}_1^2\\ =\frac{1}{2}+\frac{1}{3}+\frac{1}{2}-\frac{1}{3}-\left(2-\frac{8}{3}-\left(\frac{1}{2}-\frac{1}{3}\right)\right)\\ =-1+\frac{7}{3}+\frac{1}{2}\\ =\frac{11}{6}\end{array}$
    4. 
       $\begin{array}{l}\sqrt{x}-2=0\\ x=4\\ \\ -{\displaystyle {\int }_1^4\left(\sqrt{x}-2\right)dx}+{\displaystyle {\int }_4^9\left(\sqrt{x}-2\right)dx}\\ =-{\left[\frac{2}{3}{x}^{\frac{3}{2}}-2x\right]}_1^4+{\left[\frac{2}{3}{x}^{\frac{3}{2}}-2x\right]}_4^9\\ =-\left(\left(\frac{2}{3}·2^3-8\right)-\left(\frac{2}{3}-2\right)\right)+\left(\left(\frac{2}{3}·3^3-18\right)-\left(\frac{2}{3}·2^3-8\right)\right)\\ =-\left(\frac{16}{3}-8+\frac{4}{3}\right)-\left(\frac{16}{3}-8\right)\\ =-\frac{36}{3}+16\\ =-12+16\\ =4\end{array}$
    5. 
       $\begin{array}{l}0\le x\le \pi \\ 0\le 2x\le 2\pi \\ 2x=\pi \\ x=\frac{\pi }{2}\\ {\displaystyle {\int }_0^{\frac{\pi }{2}}\sin 2xdx}-{\displaystyle {\int }_{\frac{\pi }{2}}^{\pi }\sin 2xdx}\\ =-{\left[\frac{1}{2}\cos 2x\right]}_0^{\frac{\pi }{2}}+{\left[\frac{1}{2}\cos 2x\right]}_{\frac{\pi }{2}}^{\pi }\\ =-\frac{1}{2}\left(\cos \pi -\cos 0\right)+\frac{1}{2}\left(\cos 2\pi -\cos \pi \right)\\ =-\frac{1}{2}\left(-1-1\right)+\frac{1}{2}\left(1+1\right)\\ =1+1\\ =2\end{array}$
    6. 
       $\begin{array}{l}{e}^x=2\\ \log e^x=\log 2\\ x\log e=\log 2\\ x=\log 2\\ 0<\log 2<1\\ \\ {\displaystyle {\int }_0^{\log 2}\left(2-e^x\right)dx}+{\displaystyle {\int }_{\log 2}^1\left(e^x-2\right)dx}\\ ={\left[2x-e^x\right]}_0^{\log 2}+{\left[e^x-2x\right]}_{\log 2}^1\\ =2\log 2-e^{\log 2}-\left(-1\right)+e-2-\left(e^{\log 2}-2\log 2\right)\\ =4\log 2-2+1+e-2-2\\ =4\log 2+e-5\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, plot, Abs, sqrt, sin, pi, exp

print('22.')
x = symbols('x')
# (4) は計算が終了しなかったから省略
fs = [(Abs(x ** 2 - 1), (0, 3)),
      (Abs(x) - Abs(x - 1), (-1, 1)),
      (Abs(x - x ** 2), (-1, 2)),
      # (Abs(sqrt(x) - 2), (1, 9)),
      (Abs(sin(2 * x)), (0, pi)),
      (Abs(exp(x) - 2), (0, 1))]

# 積分の値を求められないのは、SymPy の問題か、やり方が違うからか分からず。
for i, (f, (x1, x2)) in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, (x, x1, x2))
    for o in [I, I.doit()]:
        pprint(o)
    try:
        p = plot(f, show=False, legend=True)
        p.save(f'sample22_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample22.py
22.
(1)
3            
⌠            
⎮ │ 2    │   
⎮ │x  - 1│ dx
⌡            
0            
3            
⌠            
⎮ │ 2    │   
⎮ │x  - 1│ dx
⌡            
0            

(2)
1                    
⌠                    
⎮  (│x│ - │x - 1│) dx
⌡                    
-1                   
1                    
⌠                    
⎮  (│x│ - │x - 1│) dx
⌡                    
-1                   

(3)
2             
⌠             
⎮  │ 2    │   
⎮  │x  - x│ dx
⌡             
-1            
2             
⌠             
⎮  │ 2    │   
⎮  │x  - x│ dx
⌡             
-1            

(4)
π              
⌠              
⎮ │sin(2⋅x)│ dx
⌡              
0              
π              
⌠              
⎮ │sin(2⋅x)│ dx
⌡              
0              

(5)
1            
⌠            
⎮ │ x    │   
⎮ │ℯ  - 2│ dx
⌡            
0            
1            
⌠            
⎮ │ x    │   
⎮ │ℯ  - 2│ dx
⌡            
0            

$

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="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample22.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 f4 = (x) => Math.abs(Math.sqrt(x) - 2),
    f41 = (x) => Math.sqrt(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);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[1, y1, 1, y2, 'red'],
                 [9, y1, 9, y2, 'red']],
        fns = [[f41, 'blue'],
               [f4, 'green']],
        fns1 = [],
        fns2 = [];

    fns.forEach((o) => {
        let [fn, color] = o;
        for (let x = x1; x <= x2; x += dx) {
            let y = fn(x);

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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();

(x) => Math.sqrt(x) - 2,blue
(x) => Math.abs(Math.sqrt(x) - 2),green

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