数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 回転体の体積(一般の累乗(べき乗)、逆数、0から1、極限を持つための条件、広義積分)

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15.を取り組んでみる。

$\begin{array}{l} \int_{a}^{1} π {(\frac{1}{x^{c}})}^{2} dx \\ = π \int_{a}^{1} x^{- 2 c} dx \\ = π \frac{1}{- 2 c + 1} {[x^{- 2 c + 1}]}_{a}^{1} \\ = \frac{π}{- 2 c + 1} (1 - a^{(- 2 c + 1)}) \end{array}$

極限について。

$\begin{array}{l} - 2 c + 1 > 0 \\ c < \frac{1}{2} \\ 0 < c < \frac{1}{2} \end{array}$

のとき極限をもち、その値は、

$\begin{array}{l} \lim_{a \to 0} \frac{π}{- 2 c + 1} (1 - a (- 2 c + 1)) \\ = \frac{π}{- 2 c + 1} \end{array}$

コード(Emacs)

Python 3

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

print('15.')

x = symbols('x')
c = symbols('c', positive=True)
a = symbols('a', nonnegative=True)
f = 1 / x ** c
I = Integral(pi * f ** 2, (x, a, 1))
l = Limit(I, a, 0)
for t in [I, l]:
    pprint(t)
    print()
    try:
        pprint(t.doit())
    except Exception as err:
        print(type(err), err)
    print()

for t in [l.subs({c: 1 / 3}), l.subs({c: 2 / 3})]:
    pprint(t.doit())
    print()

p = plot(f.subs({c: 1/3}), f.subs({c: 1/2}), f.subs({c: 2 / 3}),
         (x, 0, 5), ylim=(0, 5), legend=True, show=False)
colors = ['red', 'green', 'blue']

for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample15.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample15.py
15.
1           
⌠           
⎮    -2⋅c   
⎮ π⋅x     dx
⌡           
a           

⎧     -2⋅c + 1                        
⎪  π⋅a              π                 
⎪- ─────────── + ────────  for c ≠ 1/2
⎨    -2⋅c + 1    -2⋅c + 1             
⎪                                     
⎪       -π⋅log(a)           otherwise 
⎩                                     

     1           
     ⌠           
     ⎮    -2⋅c   
 lim ⎮ π⋅x     dx
a─→0⁺⌡           
     a           

<class 'NotImplementedError'> 

3.0⋅π

∞

$

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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
 <input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" 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="sample15.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 fc = (x, c) => 1 / (x ** c),
    fns = [[x => fc(x, 1 / 3), 'red'],
           [x => fc(x, 1 / 2), 'green'],
           [x => fc(x, 2 / 3), 'blue']];

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, 'brown'],
                 [1, y1, 1, 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12日水曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 回転体の体積(一般の累乗(べき乗)、逆数、0から1、極限を持つための条件、広義積分)

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