数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 面積 - 面積の公式(放物線、直線、図形の面積、比率)

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)、20.1(面積)、面積の公式、問9.を取り組んでみる。

9. 
   $\begin{array}{l}y=\sqrt{4x}=2\sqrt{x}\\ y=\frac{1}{4}{x}^2\\ 2\sqrt{x}=\frac{1}{4}{x}^2\\ 8\sqrt{x}=x^2\\ 64x=x^4\\ x^4-64x=0\\ x\left(x^3-64\right)=0\\ x=0,4\\ \\ {\displaystyle {\int }_0^4\left(2\sqrt{x}-\frac{1}{4}{x}^2\right)dx}\\ ={\left[\frac{4}{3}{x}^{\frac{3}{2}}-\frac{1}{12}{x}^3\right]}_0^4\\ =\frac{4}{3}·4^{\frac{3}{2}}-\frac{1}{12}·4^3\\ =\frac{4}{3}·{\left(2^2\right)}^{\frac{3}{2}}-\frac{16}{3}\\ =\frac{32}{3}-\frac{16}{3}\\ =\frac{16}{3}\\ \\ y=3-x\\ 2\sqrt{x}=3-x\\ 4x=9+x^2-6x\\ x^2-10x+9=0\\ \left(x-1\right)\left(x-9\right)=0\\ x=1,9\\ y\ge 0\\ 3-x\ge 0\\ x\le 3\\ x=1\\ \\ \frac{1}{4}{x}^2=3-x\\ x^2+4x-12=0\\ \left(x+6\right)\left(x-2\right)=0\\ x=-6,2\\ y\ge 0\\ 3-x\ge 0\\ x\le 3\\ x=2\\ \\ {\displaystyle {\int }_0^1\left(2\sqrt{x}-\frac{1}{4}{x}^2\right)dx}+{\displaystyle {\int }_1^2\left(\left(3-x\right)-\frac{1}{4}{x}^2\right)dx}\\ ={\displaystyle {\int }_0^{1}2\sqrt{x}dx}+{\displaystyle {\int }_0^1\left(-\frac{1}{4}{x}^2\right)dx}+{\displaystyle {\int }_1^2\left(3-x\right)dx}++{\displaystyle {\int }_1^2\left(-\frac{1}{4}{x}^2\right)dx}\\ ={\left[\frac{4}{3}{x}^{\frac{3}{2}}\right]}_0^1+{\left[3x-\frac{1}{2}{x}^2\right]}_1^2+{\displaystyle {\int }_0^2\left(-\frac{1}{4}{x}^2\right)dx}\\ =\frac{4}{3}+\left(\left(6-2\right)-\left(3-\frac{1}{2}\right)\right)-\frac{1}{4}{\left[\frac{1}{3}{x}^3\right]}_0^2\\ =\frac{4}{3}+4-\frac{5}{2}-\frac{1}{4}·\frac{8}{3}\\ =\frac{16+48-30-8}{12}\\ =\frac{26}{12}\\ =\frac{13}{6}\\ \\ \frac{13}{6}:\left(\frac{16}{3}-\frac{13}{6}\right)\\ =\frac{13}{6}:\frac{19}{6}\\ =13:19\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Rational, sqrt, solve, Integral

print('9.')
x = symbols('x')
f = 2 * sqrt(x)
g = Rational(1, 4) * x ** 2
h = 3 - x
pprint(solve(f - g))
pprint(solve(f - h))
pprint(solve(g - h))

S1 = Integral(f - g, (x, 0, 1)) + Integral(h - g, (x, 1, 2))
S2 = Integral(f - h, (x, 1, 2)) + Integral(f - g, (x, 2, 4))

for S in [S1, S2]:
    pprint(S)
    print()
    pprint(S.doit())
    print()

pprint((S1.doit() / S2.doit()).expand())

入出力結果(Terminal, Jupyter(IPython))

$ ./sample9.py
9.
[0, 4]
[1]
[-6, 2]
2                     1               
⌠                     ⌠               
⎮ ⎛   2        ⎞      ⎮ ⎛        2⎞   
⎮ ⎜  x         ⎟      ⎮ ⎜       x ⎟   
⎮ ⎜- ── - x + 3⎟ dx + ⎮ ⎜2⋅√x - ──⎟ dx
⎮ ⎝  4         ⎠      ⎮ ⎝       4 ⎠   
⌡                     ⌡               
1                     0               

13/6

                      4               
                      ⌠               
2                     ⎮ ⎛        2⎞   
⌠                     ⎮ ⎜       x ⎟   
⎮ (2⋅√x + x - 3) dx + ⎮ ⎜2⋅√x - ──⎟ dx
⌡                     ⎮ ⎝       4 ⎠   
1                     ⌡               
                      2               

19/6

13
──
19
$

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="sample9.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 f = (x) => 2 * Math.sqrt(x),
    g = (x) => 1 / 4 * x ** 2,
    h = (x) => 3 - x;

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 = [],
        fns = [[f, 'red'],
               [g, 'green']],
        fns1 = [[h, 'blue']],
        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 =