学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.3(曲線の長さ)、曲線 y = f(x) (a ≤ x ≤ b)の長さ、問26.を取り組んでみる。
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∫a0√1+(ex−e−x2)2dx=∫a0√1+e2x−2+e−2x22dx=12∫a0√22+e2x−2+e−2xdx=12∫a0√2+e2x+e−2xdx=12∫a0√(ex+e−x)2dx=12∫a0(ex+e−x)dx=12[ex−e−x]a0=12((ea−e−a)−(1−1))=ea−e−a2
∫10√1+(2x)2dxu=2xdudx=2x=0u=0x=1u=212∫20√1+u2duu=et−e−t2dudt=et+e−t2u=0t=0u=2et−e−t2=2et−e−t=4(et)2−1=4et(et)2−4et−1=0et>0et=2+√4+1=2+√5loget=log(2+√5)t=log(2+√5)12∫20√1+u2du=12∫log(2+√5)0√1+(et−e−t2)2et+e−t2dt=12∫log(2+√5)0√4+e2t+e−2t−222et+e−t2dt=123∫log(2+√5)0√e2t+e−2t+2(et+e−t)dt=123∫log(2+√5)0√(et+e−t)2(et+e−t)dt=123∫log(2+√5)0(et+e−t)(et+e−t)dt=123∫log(2+√5)0(et+e−t)2dt=123∫log(2+√5)0(e2t+2+e−2t)dt=123[12e2t+2t−12e−2t]log(2+√5)0=123((12e2log(2+√5)+2log(2+√5)−12e−2log(2+√5))−(12−12))=123(12(2+√5)2+2log(2+√5)−12(2+√5)−2)=123(12(4+5+4√5)+2log(2+√5)−12(4+5+4√5))=123(12(9+4√5)+2log(2+√5)−12(9+4√5))=123(12(9+4√5)+2log(2+√5)−9−4√52(81−16·5))=123(12(9+4√5)+2log(2+√5)−9−4√52(81−80))=123(12(9+4√5)+2log(2+√5)−9−4√52)=123(8√52+2log(2+√5))=14(2√5+log(2+√5))
∫31√1+(1x)2dx=∫31√x2+1xdxt=√x2+1dtdx=12(x2+1)−122x=x√x2+1=xtx=1t=√2x=3t=√10∫√10√2txtxdt=∫√10√2t2x2dtt=√x2+1t2=x2+1x2=t2−1∫√10√2t2x2dt=∫√10√2t2t2−1dt=∫√10√2t2−1+1t2−1dt=∫√10√2(1+1t2−1)dt=∫√10√21dt+∫√10√21t2−1dt=[t]√10√2+∫√10√21(t+1)(t−1)dt1(t+1)(t−1)=At+1+Bt−11(t+1)(t−1)=At−A+Bt+B(t+1)(t−1)1(t+1)(t−1)=(A+B)t+(B−A)(t+1)(t−1)A+B=0B−A=12B=1B=12A=−12[t]√10√2+∫√10√21(t+1)(t−1)dt=√10−√2+12∫√10√2(−1t+1+1t−1)dt=√10−√2+12[−log(t+1)+log(t−1)]√10√2=√10−√2+12((−log(√10+1)+log(√10−1))−(−log(√2+1)+log(√2−1)))=√10−√2+12log(√10−1)(√2+1)(√10+1)(√2−1)=√10−√2+12log(√10−1)2(√2+1)2(10−1)(2−1)=√10−√2+12log(√10−1)2(√2+1)29=√10−√2+12log((√10−1)(√2+1)3)2=√10−√2+log(√10−1)(√2+1)3
コード(Emacs)
Python 3
#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, Derivative, exp, log, sqrt, plot
print('26.')
x, a, b = symbols('x a b')
L = lambda f: Integral(sqrt(1 + Derivative(f, x) ** 2), (x, a, b))
fs = [((exp(x) + exp(-x)) / 2, (0, a)),
(x ** 2, (0, 1)),
(log(x), (1, 3))]
for i, (f0, (a0, b0)) in enumerate(fs, 1):
print(f'({i})')
I = L(f0).subs({a:a0, b:b0})
for t in [I, I.doit()]:
pprint(t)
print()
p = plot(f0, show=False, legend=True)
p.save(f'sample26_{i}.svg')
print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample26.py
26.
(1)
a
⌠
⎮ _____________________
⎮ ╱ 2
⎮ ╱ ⎛ ⎛ x -x⎞⎞
⎮ ╱ ⎜d ⎜ℯ ℯ ⎟⎟
⎮ ╱ ⎜──⎜── + ───⎟⎟ + 1 dx
⎮ ╲╱ ⎝dx⎝2 2 ⎠⎠
⌡
0
a
⌠
⎮ __________________
⎮ ╱ 2⋅x -2⋅x
⎮ ╲╱ ℯ + 2 + ℯ dx
⌡
0
──────────────────────────
2
(2)
1
⌠
⎮ _______________
⎮ ╱ 2
⎮ ╱ ⎛d ⎛ 2⎞⎞
⎮ ╱ ⎜──⎝x ⎠⎟ + 1 dx
⎮ ╲╱ ⎝dx ⎠
⌡
0
asinh(2) √5
──────── + ──
4 2
(3)
3
⌠
⎮ ___________________
⎮ ╱ 2
⎮ ╱ ⎛d ⎞
⎮ ╱ ⎜──(log(x))⎟ + 1 dx
⎮ ╲╱ ⎝dx ⎠
⌡
1
-√2 - asinh(1/3) + log(1 + √2) + √10
$
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="0.0001" 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="sample26.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
padding = 50,
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',
range = (start, end, step=1) => {
let res = [];
for (let i = start; i < end; i += step) {
res.push(i);
}
return res;
};
let f = (x) => 1 / 2 * (Math.exp(x) + Math.exp(-x)),
g = (x) => x ** 2,
h = (x) => Math.log(x);
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 = [[f, 'red'],
[g, 'green'],
[h, 'blue']],
fns1 = [],
fns2 = [];
fns.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = fn(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, color]);
}
}
});
fns1.forEach((o) => {
let [fn, color] = o;
lines.push([x1, fn(x1), x2, fn(x2), color]);
});
fns2.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = fn(x);
lines.push([x1, g(x1), x2, g(x2), color]);
}
});
let xscale = d3.scaleLinear()
.domain([x1, x2])
.range([padding, width - padding]);
let yscale = d3.scaleLinear()
.domain([y1, y2])
.range([height - padding, padding]);
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();
(x) => 1 / 2 * (Math.exp(x) + Math.exp(-x)),red (x) => x ** 2,green (x) => Math.log(x),blue
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