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

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18.を取り組んでみる。

$\begin{array}{l} π \int_{0}^{h} y^{2} d x \\ = π \int_{0}^{h} m^{2} x^{2} d x \\ = π m^{2} \int_{0}^{h} x^{2} d x \\ = π m^{2} {[\frac{1}{3} x^{3}]}_{0}^{h} \\ = \frac{1}{3} π m^{2} h^{3} \end{array}$
円錐の半径r = mh。

底面積。
$\begin{array}{l} π r^{2} \\ = π {(m h)}^{2} \\ = π m^{2} h^{2} \end{array}$
高さはh。
$\frac{1}{3} π m^{2} h^{3} = \frac{1}{3} π r^{2} h$
よって円錐の体積は"底面積×高さの1/3である。

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, pi

print('18.')
h, m, x = symbols('h m x')
I = pi * Integral(m ** 2 * x ** 2, (x, 0, h))

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

入出力結果(Terminal, Jupyter(IPython))

$ ./sample18.py
18.
  h         
  ⌠         
  ⎮  2  2   
π⋅⎮ m ⋅x  dx
  ⌡         
  0         

   3  2
π⋅h ⋅m 
───────
   3   

$

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="h0">h = </label>
<input id="h0" type="number" min="0" value="5">
<label for="m0">m = </label>
<input id="m0" type="number" min="0" 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="sample18.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_h0 = document.querySelector('#h0'),
    input_m0 = document.querySelector('#m0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_h0, input_m0],
    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 f1 = (x) => Math.exp(-x),
    f2 = (x) => x * f1(x),
    f3 = (x) => x ** 2 * f1(x),
    f4 = (x) => Math.log(x) / x ** 2;

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),
        h0 = parseFloat(input_h0.value),
        m0 = parseFloat(input_m0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[0, y1, 0, y2, 'red'],
                 [h0, y1, h0, y2, 'red']],
        f = (x) => m0 * x,
        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')));
    p(1 / 3 * Math.PI * m0 ** 2 * h0 ** 3);
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
h = m =

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2017年10月3日火曜日

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

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