## 2018年12月4日火曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 回転体の体積(指数関数をx軸の周りに回転、定積分、部分積分法)

1. $\begin{array}{}\int \pi {\left(\mathrm{log}x\right)}^{2}\mathrm{dx}\\ =\pi \int {\left(\mathrm{log}x\right)}^{2}\mathrm{dx}\\ =\pi \left(x{\left(\mathrm{log}x\right)}^{2}-\int x·2\left(\mathrm{log}x\right)·\frac{1}{x}\mathrm{dx}\right)\\ =\pi \left(x{\left(\mathrm{log}x\right)}^{2}-2\int \mathrm{log}x\mathrm{dx}\right)\\ =\pi \left(x{\left(\mathrm{log}x\right)}^{2}-2\left(x\mathrm{log}x-\int x·\frac{1}{x}\mathrm{dx}\right)\right)\\ =\pi \left(x{\left(\mathrm{log}x\right)}^{2}-2x\mathrm{log}x+2x\right)\\ =\pi x\left({\left(\mathrm{log}x\right)}^{2}-2\mathrm{log}x+2\right)\end{array}$

よって、求める図形を x 軸のまわりに回転してできる回転体の体積は、

$\begin{array}{}\pi {\left[x\left({\left(\mathrm{log}x\right)}^{2}-2\mathrm{log}x+2\right)\right]}_{1}^{2}\\ =\pi \left(2\left({\left(\mathrm{log}2\right)}^{2}-2\mathrm{log}2+2\right)-\left({\left(\mathrm{log}1\right)}^{2}-2\mathrm{log}1+2\right)\right)\\ =\pi \left(2{\left(\mathrm{log}2\right)}^{2}-4\mathrm{log}2+4-2\right)\\ =2\pi \left({\left(\mathrm{log}2\right)}^{2}-2\mathrm{log}2+1\right)\end{array}$

コード(Emacs)

Python 3

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

print('8.')

x = symbols('x')

f = log(x)

I = Integral(pi * f ** 2, (x, 1, 2))

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

p = plot(f, show=False, legend=True)

p.save('sample8.svg')


$./sample8.py 8. 2 ⌠ ⎮ 2 ⎮ π⋅log (x) dx ⌡ 1 ⎛ 2 ⎞ 2⋅π⋅⎝-log(4) + log (2) + 1⎠$


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="-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="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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
p = (x) => pre0.textContent += x + '\n';

let fns = [[x => Math.log(x), 'red']];

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, 'green'],
[2, y1, 2, 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();