数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 体積 - 回転体の体積(半球状の容器、鉛直線、傾き、容量)

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20章(面積、体積、長さ - 積分法の応用)、20.2(体積)、回転体の体積、問23.を取り組んでみる。

23. 
    

    半球状の容器の断面の半円を考える。

    鉛直線をx軸とし、適当にx軸、y軸、原点を定めた、半円の周の方程式。

    $x^2+y^2=r^2\left(x\ge 0\right)$

    15°傾けた後にまた15°静かに戻して、元の水の量から残った水の量を引いて、流れ出した量の水を求める。

    $\begin{array}{l}{x}^2+y^2=r^2\left(x\ge 0\right)\\ \\ {15}^{\circ }=\frac{1}{12}\pi \\ x_0=r\sin \frac{1}{12}\pi \\ =r\sin \left(\frac{3}{12}\pi -\frac{2}{12}\pi \right)\\ =r\sin \left(\frac{\pi }{4}-\frac{\pi }{6}\right)\\ =r\left(\sin \frac{\pi }{4}\cos \frac{\pi }{6}-\cos \frac{\pi }{4}\sin \frac{\pi }{6}\right)\\ =r\left(\frac{1}{\sqrt{2}}·\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}·\frac{1}{2}\right)\\ =\frac{\sqrt{3}-1}{2\sqrt{2}}r\\ =\frac{\sqrt{6}-\sqrt{2}}{4}r\\ \\ \pi {\displaystyle {\int }_0^r{y}^{2}dx}-\pi {\displaystyle {\int }_{r\sin \frac{1}{12}\pi }^r{y}^{2}dx}\\ =\pi \left({\displaystyle {\int }_0^r{y}^{2}dx}-{\displaystyle {\int }_{r\sin \frac{1}{12}\pi }^r{y}^{2}dx}\right)\\ =\pi \left({\displaystyle {\int }_0^r{y}^{2}dx}+{\displaystyle {\int }_r^{r\sin \frac{1}{12}\pi }{y}^{2}dx}\right)\\ =\pi {\displaystyle {\int }_0^{r\sin \frac{1}{12}\pi }{y}^{2}dx}\\ =\pi {\displaystyle {\int }_0^{\frac{\sqrt{6}-\sqrt{2}}{4}r}\left(r^2-x^2\right)dx}\\ =\pi {\left[r^{2}x-\frac{1}{3}{x}^3\right]}_0^{\frac{\sqrt{6}-\sqrt{2}}{4}r}\\ =\pi \left(\frac{\sqrt{6}-\sqrt{2}}{4}{r}^3-\frac{1}{3}{\left(\frac{\sqrt{6}-\sqrt{2}}{4}r\right)}^3\right)\\ =\pi \left(\frac{\sqrt{6}-\sqrt{2}}{4}-\frac{6\sqrt{6}-3·6\sqrt{2}+3\sqrt{6}·2-2\sqrt{2}}{3·4^3}\right)r^3\\ =\pi \left(\frac{\sqrt{6}-\sqrt{2}}{4}-\frac{12\sqrt{6}-20\sqrt{2}}{3·4^3}\right)r^3\\ =\pi \left(\frac{\sqrt{6}-\sqrt{2}}{4}-\frac{3\sqrt{6}-5\sqrt{2}}{3·4^2}\right)r^3\\ =\frac{12\sqrt{6}-12\sqrt{2}-3\sqrt{6}+5\sqrt{2}}{3·4^2}\pi r^3\\ =\frac{9\sqrt{6}-7\sqrt{2}}{3·4^2}\pi r^3\\ =\frac{9\sqrt{6}-7\sqrt{2}}{48}\pi r^3\left(c{m}^3\right)\end{array}$

コード(Emacs)

Python 3

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

print('23.')
x, r = symbols('x r')
y2 = r ** 2 - x ** 2
V = pi * Integral(y2, (x, 0, r)) - pi * Integral(y2, (x, r * sin(pi / 12), r))

for t in [V, V.doit().factor()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample23.py
23.
  r                        r                   
  ⌠                        ⌠                   
  ⎮ ⎛ 2    2⎞              ⎮       ⎛ 2    2⎞   
π⋅⎮ ⎝r  - x ⎠ dx - π⋅      ⎮       ⎝r  - x ⎠ dx
  ⌡                        ⌡                   
  0                    ⎛  √2   √6⎞             
                     r⋅⎜- ── + ──⎟             
                       ⎝  4    4 ⎠             

    3                
-π⋅r ⋅(-9⋅√6 + 7⋅√2) 
─────────────────────
          48         

$

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">
<br>
<label for="r1">r1 = </label>
<input id="r1" type="number" value="5">

<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="sample23.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_r1 = document.querySelector('#r1'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_r1],
    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 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),
        r1 = parseFloat(input_r1.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[x1, -r1 * Math.sin(Math.PI / 12),
                  x2, -r1 * Math.sin(Math.PI / 12), 'red']],
        f = (x) => -Math.sqrt(r1 ** 2 - x ** 2),
        fns = [[f, '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();

r = dx =
x1 = x2 =
y1 = y2 =
r1 =