## 2018年11月10日土曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 曲線の長さ(極座標表示、三角関数(正弦、余弦)、輪の長さ、置換積分法)

1. $\begin{array}{}\int \sqrt{{\left(1+\mathrm{cos}\theta \right)}^{2}+{\left(\frac{d}{d\theta }\left(1+\mathrm{cos}\theta \right)\right)}^{2}}d\theta \\ =\int \sqrt{1+2\mathrm{cos}\theta +{\mathrm{cos}}^{2}\theta +{\left(-\mathrm{sin}\theta \right)}^{2}}d\theta \\ =\int \sqrt{1+2\mathrm{cos}\theta +{\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta }d\theta \\ =\int \sqrt{2+2\mathrm{cos}\theta }d\theta \\ =\sqrt{2}\int \sqrt{1+\mathrm{cos}\theta }d\theta \\ t=\mathrm{cos}\theta \\ 1=-\mathrm{sin}\theta \frac{d\theta }{\mathrm{dt}}\\ d\theta =-\frac{1}{\mathrm{sin}\theta }\mathrm{dt}\\ {\mathrm{sin}}^{2}\theta =1-{\mathrm{cos}}^{2}\theta =1-{t}^{2}\\ \underset{0}{\overset{2\pi }{\int }}\sqrt{1+\mathrm{cos}\theta }d\theta \\ =\underset{1}{\overset{-1}{\int }}\sqrt{1+t}·\left(-\frac{1}{\sqrt{1-{t}^{2}}}\right)\mathrm{dt}+\underset{-1}{\overset{1}{\int }}\sqrt{1+t}·\frac{1}{\sqrt{1-{t}^{2}}}\mathrm{dt}\\ =\underset{-1}{\overset{1}{\int }}\frac{1}{\sqrt{1-t}}\mathrm{dt}+{\int }_{-1}^{1}\frac{1}{\sqrt{1-t}}\mathrm{dt}\\ =2{\left[-2\sqrt{1-t}\right]}_{-1}^{1}\\ =-4\left(0-\sqrt{2}\right)\\ =4\sqrt{2}\end{array}$

よって求める1つの輪の長さは、

$\sqrt{2}4\sqrt{2}=8$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, Derivative, sqrt, cos, pi

print('6.')
θ = symbols('θ')
f = 1 + cos(θ)
I = Integral(sqrt(f ** 2 + Derivative(f, θ, 1) ** 2), (θ, 0, 2 * pi))

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


$./sample8.py 6. 2⋅π ⌠ ⎮ ___________________________________ ⎮ ╱ 2 ⎮ ╱ 2 ⎛d ⎞ ⎮ ╱ (cos(θ) + 1) + ⎜──(cos(θ) + 1)⎟ dθ ⎮ ╲╱ ⎝dθ ⎠ ⌡ 0 2⋅π ⌠ ⎮ ______________ ⎮ ╲╱ 2⋅cos(θ) + 2 dθ ⌡ 0$


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="-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">

<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="sample8.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'),
input_n0 = document.querySelector('#n0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
p = (x) => pre0.textContent += x + '\n';

let fr = theta => 1 + Math.cos(theta),
fx = theta => fr(theta) * Math.cos(theta),
fy = theta => fr(theta) * Math.sin(theta),
fns = [];

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 = [];

for (let theta = 0; theta <= 2 * Math.PI; theta += dx) {
points.push([fx(theta), fy(theta), 'green']);
}

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();