数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 面積 - 広義の積分(極限、無限区間, 累乗(べき乗)、指数関数、対数関数)

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)、20.1(面積)、広義の積分、問16.を取り組んでみる。

1. $\begin{array}{l} b > 0 \\ \lim_{b \to \infty} \int_{0}^{b} e^{- x} d x \\ = \lim_{b \to \infty} {[- e^{- x}]}_{0}^{b} \\ = \lim_{b \to \infty} (- e^{- b} + e^{- 0}) \\ = \lim_{b \to \infty} (- e^{- b}) + \lim_{b \to \infty} 1 \\ = 0 + 1 \\ = 1 \end{array}$
2. $\begin{array}{l} b > 0 \\ \lim_{b \to \infty} \int_{0}^{b} x e^{- x} d x \\ = \lim_{b \to \infty} ({[- x e^{- x}]}_{0}^{b} - \int_{0}^{b} (- e^{- x}) d x) \\ = \lim_{b \to \infty} ((- b e^{- b} + 0) + \int_{0}^{b} e^{- x} d x) \\ = \lim_{b \to \infty} (- \frac{b}{e^{b}}) + \lim_{b \to \infty} (\int_{0}^{b} e^{- x} d x) \\ = 0 + 1 \\ = 1 \end{array}$
3. $\begin{array}{l} b > 0 \\ \lim_{b \to \infty} \int_{0}^{b} x^{2} e^{- x} d x \\ = \lim_{b \to \infty} ({[- x^{2} e^{- x}]}_{0}^{b} - \int_{0}^{b} (- 2 x e^{- x}) d x) \\ = \lim_{b \to \infty} ((- b^{2} e^{- b} + 0) + 2 \int_{0}^{b} x e^{- x} d x) \\ = \lim_{b \to \infty} (- \frac{b^{2}}{e^{b}}) + \underset{b \to \infty}{2 \lim} (\int_{0}^{b} x e^{- x} d x) \\ = 0 + 2 \cdot 1 \\ = 2 \end{array}$
4. $\begin{array}{l} b > 1 \\ \lim_{b \to \infty} \int_{1}^{b} \frac{\log x}{x^{2}} d x \\ = \lim_{b \to \infty} ({[- x^{- 1} \log x]}_{1}^{b} - \int_{1}^{b} (- \frac{x^{- 1}}{x}) d x) \\ = \lim_{b \to \infty} ((- b^{- 1} \log b + 1 \log 1) + \int_{1}^{b} x^{- 2} d x) \\ = \lim_{b \to \infty} ((- b^{- 1} \log b + 0) + {[- x^{- 1}]}_{1}^{b}) \\ = \lim_{b \to \infty} (- \frac{\log b}{b}) - \lim_{b \to \infty} ({[x^{- 1}]}_{1}^{b}) \\ = 0 - \lim_{b \to \infty} (b^{- 1} - 1) \\ = 1 \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral, oo, exp, log

print('16.')
x = symbols('x')
fs = [(exp(-x), (0, oo)),
      (x * exp(-x), (0, oo)),
      (x ** 2 * exp(-x), (0, oo)),
      (log(x) / x ** 2, (1, oo))]

for i, (f, (x1, x2)) in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, (x, x1, x2))
    for t in [I, I.doit()]:
        pprint(t)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample16.py
16.
(1)
∞       
⌠       
⎮  -x   
⎮ ℯ   dx
⌡       
0       

1


(2)
∞         
⌠         
⎮    -x   
⎮ x⋅ℯ   dx
⌡         
0         

1


(3)
∞          
⌠          
⎮  2  -x   
⎮ x ⋅ℯ   dx
⌡          
0          

2


(4)
∞          
⌠          
⎮ log(x)   
⎮ ────── dx
⎮    2     
⎮   x      
⌡          
1          

1


$

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="-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="sample16.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 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);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[0, y1, 0, y2, 'red'],
                 [1, y1, 1, y2, 'red']],
        fns = [[f1, 'green'],
               [f2, 'blue'],
               [f3, 'orange'],
               [f4, 'brown']],
        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(h(m0));
};

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

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2017年10月1日日曜日

数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 面積 - 広義の積分(極限、無限区間, 累乗(べき乗)、指数関数、対数関数)

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