## 2019年2月4日月曜日

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

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

コード

Python 3

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

theta = symbols('θ')
r = sqrt(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('sample3.png')


$python3 sample3.py 2⋅π ⌠ ⎮ ⎛ cos(θ) 1⎞ ⎮ ⎜- ────── + ─⎟ dθ ⎮ ⎝ 2 2⎠ ⌡ 0 π$