数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 回転体の体積(2次関数と直線に挟まれる領域、x軸の周りに回転、定積分)

学習環境

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)、3(回転体の体積)の練習問題5.を取り組んでみる。

$\begin{array}{l} x^{2} = 5 x = 0 \\ x^{2} - 5 x = 0 \\ x (x - 5) = 0 \\ x = 0, 5 \\ 0 \leq x \leq 5 \\ 5 x - x^{2} \geq 0 \end{array}$

よって、求める2次関数と直線にはさまれる領域を x軸のまわりに回転してできる回転体の体積は、

$\begin{array}{l} \int_{0}^{5} π {(5 x)}^{2} dx - \int_{0}^{5} π {(x^{2})}^{2} dx \\ = π \int_{0}^{5} (5^{2} x^{2} - x^{4}) dx \\ = π {[\frac{5^{2}}{3} x^{3} - \frac{1}{5} x^{5}]}_{0}^{5} \\ = π (\frac{5^{2}}{3} \cdot 5^{3} - \frac{1}{5} \cdot 5^{5}) \\ = \frac{π (5^{5} - 3 \cdot 5^{4})}{3} \\ = \frac{2 \cdot 5^{4} π}{3} \end{array}$

コード(Emacs)

Python 3

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

print('5.')
x = symbols('x')
f = 5 * x
g = x ** 2
x1, x2 = sorted(solve(f - g, x))

I = Integral(pi * f ** 2, (x, x1, x2)) - Integral(pi * g ** 2, (x, x1, x2))

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

p = plot(f, g, show=False, legend=True)
colors = ['red', 'green']
for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample5.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
5.
  5           5           
  ⌠           ⌠           
  ⎮    4      ⎮       2   
- ⎮ π⋅x  dx + ⎮ 25⋅π⋅x  dx
  ⌡           ⌡           
  0           0           

1250⋅π
──────
  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="-25">
<label for="x2">x2 = </label>
 <input id="x2" type="number" value="25">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-25">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="25">

<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="sample5.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';

let fns = [[x => x ** 2, 'red'],
           [x => 5 * x, 'green']];

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 = [[0, y1, 0, y2, 'blue'],
                 [5, y1, 5, y2, 'orange']];

    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 =

Kamimura's blog

2018年12月1日土曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 回転体の体積(2次関数と直線に挟まれる領域、x軸の周りに回転、定積分)

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