## 2018年9月20日木曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数に関する積分(部分積分方、累乗、平方根、微分、不定積分)

1. $\begin{array}{}\int \frac{{x}^{3}}{\sqrt{16-{x}^{2}}}\mathrm{dx}\\ =\left(-{x}^{2}\right)\sqrt{16-{x}^{2}}+2\int x\sqrt{16-{x}^{2}}\mathrm{dx}\\ \int x\sqrt{16-{x}^{2}}\mathrm{dx}\\ =-\frac{1}{3}{\left(16-{x}^{2}\right)}^{\frac{3}{2}}\\ -{x}^{2}\sqrt{16-{x}^{2}}-\frac{2}{3}{\left(16-{x}^{2}\right)}^{\frac{3}{2}}\end{array}$

コード(Emacs)

Python 3

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

print('15.')

x = symbols('x')
f = x ** 3 / sqrt(16 - x ** 2)
I = Integral(f, x)
for t in [I, I.doit().simplify()]:
pprint(t)
print()

p = plot(f, I.doit(), legend=True, show=False)
colors = ['red', 'green']
for i, c in enumerate(colors):
p[i].line_color = c
p.save('sample15.svg')

g = -x ** 2 * sqrt(16 - x ** 2) - Rational(2, 3) * \
(16 - x ** 2) ** Rational(3, 2)

Dg = Derivative(g, x, 1)
for t in [Dg, Dg.doit()]:
pprint(t)
print()


$./sample15.py 15. ⌠ ⎮ 3 ⎮ x ⎮ ────────────── dx ⎮ ___________ ⎮ ╱ 2 ⎮ ╲╱ - x + 16 ⌡ ___________ ╱ 2 ⎛ 2 ⎞ -╲╱ - x + 16 ⋅⎝x + 32⎠ ────────────────────────── 3 ⎛ 3/2⎞ ⎜ ___________ ⎛ 2 ⎞ ⎟ d ⎜ 2 ╱ 2 2⋅⎝- x + 16⎠ ⎟ ──⎜- x ⋅╲╱ - x + 16 - ────────────────⎟ dx⎝ 3 ⎠ 3 x ────────────── ___________ ╱ 2 ╲╱ - x + 16$


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">

<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'),
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 f = (x) => x ** 3 / Math.sqrt(16 - x ** 2),
g = (x) => -1 / 3 * Math.sqrt(16 - x ** 2) * (x ** 2 + 32),
fns = [[f, 'red'],
[g, '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 = [];

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();