数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 曲線の長さ(パラメーター、三角関数(正弦と余弦)の累乗(立方)、加法定理、2倍角)

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4.を取り組んでみる。

4. 
   
   1. 
      
      $\begin{array}{}\cos t\ge \; <!--\; greater-than\; or\; equal\; to\; -->0,\sin t\ge \; <!--\; greater-than\; or\; equal\; to\; -->0,0\le \; <!--\; less-than\; or\; equal\; to\; -->t\le \; <!--\; less-than\; or\; equal\; to\; -->\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}\\ \frac{\mathrm{dx}}{\mathrm{dt}}=-3{\cos }^{2}t\sin t\\ \frac{\mathrm{dy}}{\mathrm{dt}}=3{\sin }^{2}t\cos t\\ \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^2+{\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^2}\\ =\sqrt{3^2{\cos }^{4}t{\sin }^{2}t+3^2{\sin }^{4}t{\cos }^{2}t}\\ =3\cos t\sin t\sqrt{{\cos }^{2}t+{\sin }^{2}t}\\ =3\cos t\sin t\\ 3\int \; <!--\; integral\; -->\cos t\sin t\mathrm{dt}\\ =3\int \; <!--\; integral\; -->\frac{\sin \left(2t\right)}{2}\mathrm{dt}\\ =-\frac{3}{4}\cos \left(2t\right)\\ -\frac{3}{4}{\left[\cos \left(2t\right)\right]}_0^{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{4}}\\ =-\frac{3}{4}\left(0-1\right)\\ =\frac{3}{4}\end{array}$
   2. 
      

      a の場合の平方根の部分を三角関数の正負(t の値の範囲による）に合わせて修正。

      $\begin{array}{}0\le \; <!--\; less-than\; or\; equal\; to\; -->t\le \; <!--\; less-than\; or\; equal\; to\; -->\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2},\cos t\ge \; <!--\; greater-than\; or\; equal\; to\; -->0,\sin t\ge \; <!--\; greater-than\; or\; equal\; to\; -->0\\ \frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}\le \; <!--\; less-than\; or\; equal\; to\; -->t\le \; <!--\; less-than\; or\; equal\; to\; -->\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->},\cos t\le \; <!--\; less-than\; or\; equal\; to\; -->0,\sin t\ge \; <!--\; greater-than\; or\; equal\; to\; -->0\\ \underset{0}{\overset{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{\int \; <!--\; integral\; -->}}\sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^2+{\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^2}\mathrm{dt}\\ =\underset{-0}{\overset{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}}{\int \; <!--\; integral\; -->}}\sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^2+{\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^2}\mathrm{dt}+\underset{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}}{\overset{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{\int \; <!--\; integral\; -->}}\sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^2+{\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^2}\mathrm{dt}\\ =-\frac{3}{4}{\left[\cos \left(2t\right)\right]}_0^{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}}+\frac{3}{4}{\left[\cos \left(2t\right)\right]}_{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}\\ =\frac{3}{4}\left(-\left(-1-1\right)+\left(1+1\right)\right)\\ =3\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, sqrt, Derivative, pi, sin, cos

print('4.')
t, a, b = symbols('t, a, b', real=True)
intervals = [(0, pi / 4), (0, pi)]
I = Integral(
    sqrt(Derivative(cos(t) ** 3, t, 1) ** 2 +
         Derivative(sin(t) ** 3, t, 1) ** 2),
    (t, a, b))

for i, (t1, t2) in enumerate(intervals):
    print(f'({chr(ord("a") + i)})')
    pprint(I.subs({a: t1, b: t2}).doit().simplify())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
(a)
3/4

(b)
3

$

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="-2">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="2">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-2">
<label for="y2">y2 = </label>
<input id="y2" type="number" 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="sample4.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 fx = t => Math.cos(t) ** 3,
    fy = t => Math.sin(t) ** 3,
    fns = [];

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 = [];

    for (let t = 0; t <= Math.PI / 4; t += dx) {
        points.push([fx(t), fy(t), 'green']);
    }
    for (let t = Math.PI / 4; t <= Math.PI; t += dx) {
        points.push([fx(t), fy(t), '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 =