## 2019年2月3日日曜日

### 数学 - Python - 解析学 - 積分 - 積分の応用 - 面積(三角関数(余弦)、累乗、極座標表示)

1. $\begin{array}{}\underset{0}{\overset{2\pi }{\int }}\frac{1}{2\pi }·\pi {r}^{2}d\theta \\ =\underset{0}{\overset{2\pi }{\int }}\frac{1}{2}{r}^{2}d\theta \\ =\frac{1}{2}\underset{0}{\overset{2\pi }{\int }}{\left(1-\mathrm{cos}\theta \right)}^{2}d\theta \\ =\frac{1}{2}\underset{0}{\overset{2\pi }{\int }}\left(1-2\mathrm{cos}\theta +{\mathrm{cos}}^{2}\theta \right)d\theta \\ =\frac{1}{2}\left({\left[\theta -2\mathrm{sin}\theta \right]}_{0}^{2\pi }+\frac{1}{2}{\left[\mathrm{cos}\theta \mathrm{sin}\theta \right]}_{0}^{2\pi }+\frac{1}{2}\underset{0}{\overset{2\pi }{\int }}1d\theta \right)\\ =\frac{1}{2}\left(2\pi +\frac{1}{2}·2\pi \right)\\ =\frac{\pi }{2}\left(2+1\right)\\ =\frac{3}{2}\pi \end{array}$

コード

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, cos, sin, pi, exp
from sympy.plotting import plot_parametric

theta = symbols('θ')
r = 1 - cos(theta)
x = r * cos(theta)
y = r * sin(theta)

I = Integral(r ** 2 / 2, (theta, 0, 2 * pi))

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

p = plot_parametric((x, y, (theta, 0, pi / 2)),
(x, y, (theta, pi / 2, pi)),
(x, y, (theta, pi, 3 * pi / 2)),
(x, y, (theta, 3 * pi / 2, 2 * pi)),
show=False)

colors = ['red', 'green', 'blue', 'brown']
for i, s in enumerate(p):
s.line_color = colors[i]
p.save('sample2.png')

$python3 sample2.py 2⋅π ⌠ ⎮ 2 ⎮ (cos(θ) - 1) ⎮ ───────────── dθ ⎮ 2 ⌡ 0 3⋅π ─── 2$