2018年5月31日木曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(円の面積、正弦、余弦、2倍角、部分積分法、平方根)

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

コード(Emacs)

Python 3

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

x, y = symbols('x, y', real=True)
r = symbols('r', positive=True)
eq = x ** 2 + y ** 2 - r ** 2
ys = solve(eq, y)
y = ys[1]
I = 4 * Integral(y, (x, 0, r))
for t in [I, I.doit()]:
pprint(t)
print()

p = plot(*[y0.subs({r: 1}) for y0 in ys], legend=True, show=False)
for i, color in enumerate(['red', 'green']):
p[i].line_color = color
p.save('sample7.svg')

for t in ys:
pprint(t)
print()


$./sample7.py r ⌠ ⎮ _________________ 4⋅⎮ ╲╱ (r - x)⋅(r + x) dx ⌡ 0 2 π⋅r _________________ -╲╱ (r - x)⋅(r + x) _________________ ╲╱ (r - x)⋅(r + x)$


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="0.001" 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">
<br>
<label for="r1">r1 = </label>
<input id="r1" type="number" value="2">

<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="sample7.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_r1 = document.querySelector('#r1'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_r1],
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 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),
r1 = parseFloat(input_r1.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
lines = [[r1, y1, r1, y2, 'blue'],
[-r1, y1, -r1, y2, 'orange'],
[x1, r1, x2, r1, 'brown'],
[x1, -r1, x2, -r1, 'purple'] ],
fns = [[(x) => Math.sqrt(r1 ** 2 - x ** 2), 'red'],
[(x) => -Math.sqrt(r1 ** 2 - x ** 2), 'green']],
fns1 = [],
fns2 = [];

fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);

points.push([x, y, color]);
}
});

fns1
.forEach((o) => {
let [f, color] = o;

lines.push([x1, f(x1), x2, f(x2), color]);
});

fns2
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = f(x);
lines.push([x1, g(x1), x2, g(x2), 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, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

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