## 2018年11月21日水曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 極座標による面積(三角関数(余弦)、3倍角、累乗(べき乗)、定積分)

1. $\begin{array}{}\int \frac{1}{2}{\left(\mathrm{cos}\left(3\theta \right)\right)}^{2}d\theta \\ =\frac{1}{2}\int \mathrm{cos}{\left(3\theta \right)}^{2}d\theta \\ t=3\theta \\ \frac{\mathrm{dt}}{d\theta }=3\\ \frac{1}{2}\int {\mathrm{cos}}^{2}t·\frac{1}{3}\mathrm{dt}\\ =\frac{1}{6}\left(\mathrm{cos}t\mathrm{sin}t+\frac{1}{2}\int 1\mathrm{dt}\right)\\ =\frac{1}{12}\left(2\mathrm{cos}t\mathrm{sin}t+t\right)\\ \theta =±\frac{\pi }{6},t=±\frac{\pi }{2}\end{array}$

よって、求める曲線で囲まれる図形の面積は、

$\begin{array}{}\frac{1}{12}\left[2\mathrm{cos}t\mathrm{sin}t+t\right]-\frac{\frac{\pi }{{\pi }^{2}}}{2}\\ =\frac{1}{12}\left(\frac{\pi }{2}+\frac{\pi }{2}\right)\\ =\frac{\pi }{12}\end{array}$

コード(Emacs)

Python 3

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

print('3.')
θ = symbols('θ')
f = cos(3 * θ)
I = Integral(Rational(1, 2) * f ** 2, (θ, -pi / 6, pi / 6))

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


$./sample4.py 3. π ─ 6 ⌠ ⎮ 2 ⎮ cos (3⋅θ) ⎮ ───────── dθ ⎮ 2 ⌡ -π ─── 6 π ── 12$


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="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1.5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1.5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.5">

<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="sample4.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 fr = theta => Math.cos(3 * 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 = -Math.PI / 6; theta < Math.PI / 6; theta += dx) {
points.push([fx(theta), fy(theta), 'red']);
}
for (let theta = Math.PI / 6; theta < 3 / 6 * Math.PI; theta += dx) {
points.push([fx(theta), fy(theta), 'green']);
}
for (let theta = 3 / 6 * Math.PI; theta < 5 / 6 * Math.PI; theta += dx) {
points.push([fx(theta), fy(theta), '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();