## 2019年2月11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)、極座標表示)

1. $\begin{array}{}\int \frac{\pi {r}^{2}}{2\pi }d\theta \\ =\frac{1}{2}\int {\left(2+\mathrm{cos}\theta \right)}^{2}d\theta \\ =\frac{1}{2}\int \left(4+4\mathrm{cos}\theta +{\mathrm{cos}}^{2}\theta \right)d\theta \\ =\frac{1}{2}\left(4\theta +4\mathrm{sin}\theta +\frac{1}{2}\mathrm{cos}\theta \mathrm{sin}\theta +\frac{1}{2}\int 1d\theta \right)\\ =2\theta +2\mathrm{sin}\theta +\frac{1}{4}\left(\mathrm{cos}\theta \mathrm{sin}\theta +\theta \right)\\ {\left[2\theta +2\mathrm{sin}\theta +\frac{1}{4}\left(\mathrm{cos}\theta \mathrm{sin}\theta +\theta \right)\right]}_{0}^{2\pi }\\ =4\pi +\frac{\pi }{2}\\ =\frac{9}{2}\pi \end{array}$

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 = 2 + 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', 'orange', 'pink']
for i, s in enumerate(p):
s.line_color = colors[i]
p.save('sample10.png')


C:\Users\...> py -3 sample10.py
2⋅π
⌠
⎮              2
⎮  (cos(θ) + 2)
⎮  ───────────── dθ
⎮        2
⌡
0

9⋅π
───
2

C:\Users\...>