数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 体積 - 回転体の体積(扇形、x軸)

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

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22.を取り組んでみる。

扇形一辺をx軸とする。

原点を中心とした半径rの円の方程式。
$\begin{array}{l} x^{2} + y^{2} = r^{2} \\ y^{2} = r^{2} - x^{2} \end{array}$
yが非負の部分の方程式。
$y = \sqrt{r^{2} - x^{2}}$
半円をx軸のまわりに回転してできる回転体の体積から不要な部分の体積を除いて求める扇形の回転体の体積を求める。(除く回転体の体積は中心角60°の扇形をx軸のまわりに回転してできる回転体の体積。)
$\begin{array}{l} x^{2} + y^{2} = r^{2} \\ y^{2} = r^{2} - x^{2} \\ 120^{\circ} = \frac{2}{3} π \\ V = π \int_{- r}^{r} y^{2} d x - (π \int_{- r}^{r \cos \frac{2}{3} π} y^{2} d x + π \int_{r \cos \frac{2}{3} π}^{0} {(x \tan \frac{2}{3} π)}^{2} d x) \\ = π (\int_{- r}^{r} y^{2} d x - (\int_{- r}^{r \cos \frac{2}{3} π} y^{2} d x + \int_{r \cos \frac{2}{3} π}^{0} {(x \tan \frac{2}{3} π)}^{2} d x)) \\ = π (\int_{- r}^{r} (r^{2} - x^{2}) d x - \int_{- r}^{- \frac{r}{2}} (r^{2} - x^{2}) d x - \int_{- \frac{r}{2}}^{0} {(- \sqrt{3} x)}^{2} d x) \\ = π (2 \int_{0}^{r} (r^{2} - x^{2}) d x - {[r^{2} x - \frac{1}{3} x^{3}]}_{- r}^{- \frac{r}{2}} - \int_{- \frac{r}{2}}^{0} 3 x^{2} d x) \\ = π (2 {[r^{2} x - \frac{1}{3} x^{3}]}_{0}^{r} - ((- \frac{r^{3}}{2} - \frac{1}{3} (- \frac{r^{3}}{8})) - (- r^{3} + \frac{r^{3}}{3})) - {[x^{3}]}_{- \frac{r}{2}}^{0}) \\ = π (2 (r^{3} - \frac{r^{3}}{3}) - (- \frac{r^{3}}{2} + \frac{r^{3}}{24} + \frac{2 r^{3}}{3}) - (- (- \frac{r^{3}}{8}))) \\ = π (\frac{4 r^{3}}{3} + \frac{r^{3}}{2} - \frac{r^{3}}{24} - \frac{2 r^{3}}{3} - \frac{r^{3}}{8}) \\ = π \cdot \frac{32 + 12 - 1 - 16 - 3}{24} r^{3} \\ = π r^{3} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sqrt, cos, tan, Integral, pi

print('22.')
x, r = symbols('x r')
y = sqrt(r ** 2 - x ** 2)
V = pi * Integral(y ** 2, (x, -r, r)) - (
    pi * Integral(y ** 2, (x, -r, r * cos(2 * pi / 3))) +
    pi * Integral((x * tan(2 * pi / 3)) ** 2, (x, r * cos(2 * pi / 3), 0))
)


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

入出力結果(Terminal, Jupyter(IPython))

$ ./sample22.py
22.
                    -r                                  
                    ───                                 
     0               2                   r              
     ⌠               ⌠                   ⌠              
     ⎮     2         ⎮  ⎛ 2    2⎞        ⎮  ⎛ 2    2⎞   
- π⋅ ⎮  3⋅x  dx - π⋅ ⎮  ⎝r  - x ⎠ dx + π⋅⎮  ⎝r  - x ⎠ dx
     ⌡               ⌡                   ⌡              
    -r               -r                  -r             
    ───                                                 
     2                                                  

   3
π⋅r 

$

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="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'),
    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 = [[r1 * Math.cos(2 / 3 * Math.PI), y1,
                  r1 * Math.cos(2 / 3 * Math.PI), y2, 'red']],
        f = (x) => Math.sqrt(r1 ** 2 - x ** 2),
        fns = [[f, 'green']],
        fns1 = [[(x) => x * Math.tan(2 / 3 * Math.PI), '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 =
r1 =

Kamimura's blog

2017年10月7日土曜日

数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 体積 - 回転体の体積(扇形、x軸)

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