数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 曲線の長さ(累乗関数(べき乗)、置換積分法)

解析入門 原書第3版
原著
楽天ブックス(Kobo) Yahoo!

学習環境

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

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13章(積分の応用)、1(曲線の長さ)の練習問題6.を取り組んでみる。

6. 
   
   $\begin{array}{}\int \; <!--\; integral\; -->\sqrt{1+{\left(\frac{d}{\mathrm{dx}}{x}^{\frac{3}{2}}\right)}^2}\mathrm{dx}\\ =\int \; <!--\; integral\; -->\sqrt{1+{\left(\frac{3}{2}{x}^{\frac{1}{2}}\right)}^2}\mathrm{dx}\\ =\int \; <!--\; integral\; -->\sqrt{1+\frac{9}{4}x}\mathrm{dx}\\ =\frac{1}{2}\int \; <!--\; integral\; -->\sqrt{4+9x}\mathrm{dx}\\ t^2=4+9x\\ 2t=9·\; <!--\; middle\; dot\; -->\frac{\mathrm{dx}}{\mathrm{dt}}\\ \mathrm{dx}=\frac{2}{9}t\mathrm{dt}\\ \frac{1}{2}\int \; <!--\; integral\; -->t·\; <!--\; middle\; dot\; -->\frac{2}{9}t\mathrm{dt}\\ =\frac{1}{9}\int \; <!--\; integral\; -->t^2\mathrm{dt}\\ =\frac{1}{27}{t}^3\\ x=1,t=\sqrt{13}\\ x=3,t=\sqrt{31}\\ \frac{1}{27}\left(31\sqrt{31}-13\sqrt{13}\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, Derivative, sqrt, plot, Rational

print('6.')
x = symbols('x')
f = x ** Rational(3, 2)
I = Integral(sqrt(1 + Derivative(f, x, 1) ** 2), (x, 1, 3))

for t in [I, I.doit()]:
    pprint(t.simplify())
    print()

p = plot(f, show=False, legend=True)
p.save('sample6.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
6.
3                          
⌠                          
⎮      _________________   
⎮     ╱           2        
⎮    ╱  ⎛d ⎛ 3/2⎞⎞         
⎮   ╱   ⎜──⎝x   ⎠⎟  + 1  dx
⎮ ╲╱    ⎝dx      ⎠         
⌡                          
1                          

  13⋅√13   31⋅√31
- ────── + ──────
    27       27  

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.01">
<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="sample6.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],
    p = (x) => pre0.textContent += x + '\n';

let fns = [[(x) => Math.sqrt(x ** 3), 'red']];

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, 'green'],
                 [3, y1, 3, y2, 'blue']];
    
    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, 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].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 =