## 2018年12月11日火曜日

### 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、極限を持つための条件、広義積分)

1. $\begin{array}{}\underset{1}{\overset{B}{\int }}\pi {y}^{2}\mathrm{dx}\\ =\pi \underset{1}{\overset{\beta }{\int }}{\left(\frac{1}{{x}^{c}}\right)}^{2}\mathrm{dx}\\ =\pi {\int }_{1}^{B}{x}^{-2c}\mathrm{dx}\\ =\pi {\left[\frac{1}{-2c+1}{x}^{-2c+1}\right]}_{1}^{B}\\ =\frac{\pi }{-2c+1}\left({B}^{-2c+1}-1\right)\\ =\frac{\pi }{2c-1}\left(1-{B}^{-2c+1}\right)\end{array}$

極限について。

$\begin{array}{}\underset{B\to \infty }{\mathrm{lim}}\frac{\pi }{2c-1}\left(1-{B}^{-2c+1}\right)\\ =\frac{\pi }{2c-1}-\underset{B\to \infty }{\mathrm{lim}}{B}^{-2c+1}\end{array}$

よって、

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

ならば極限をもち、その値は、

$\frac{\pi }{2c-1}$

である。

コード(Emacs)

Python 3

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

print('14.')

x = symbols('x')
c = symbols('c', positive=True)
B = symbols('B', nonnegative=True)
f = 1 / x ** c
I = Integral(pi * f ** 2, (x, 1, B))
l = Limit(I, B, oo)
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: 1})]:
pprint(t.doit())
print()

p = plot(f.subs({c: 1/3}), f.subs({c: 1}),
ylim=(-10, 10), legend=True, show=False)
colors = ['red', 'green']

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

p.save('sample14.png')


$./sample14.py 14. B ⌠ ⎮ -2⋅c ⎮ π⋅x dx ⌡ 1 ⎧ -2⋅c + 1 ⎪π⋅B π ⎪─────────── - ──────── for c ≠ 1/2 ⎨ -2⋅c + 1 -2⋅c + 1 ⎪ ⎪ π⋅log(B) otherwise ⎩ B ⌠ ⎮ -2⋅c lim ⎮ π⋅x dx B─→∞⌡ 1 <class 'NotImplementedError'> ∞ π$


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="sample14.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
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), '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 = [[1, y1, 1, y2, 'blue']];

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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();