3重積分重心密度、一様、物体、三角関数(正弦と余弦)、累乗、加法定理、倍角

続解析入門 (原書第2版)
楽天ブックス
Yahoo!ショッピング
au PAY マーケット
原著: Calculus of Several Variables

学習環境

続解析入門 (原書第2版) (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第9章(3重積分)、3(重心)の練習問題6の解答を求めてみる。

\begin{array}{l} m \\ = \int_{0}^{\frac{π}{4}} (\cos x - \sin x) dx \\ = {[\sin x + \cos x]}_{0}^{\frac{π}{4}} \\ = \frac{2}{\sqrt{2}} - 1 \\ = \sqrt{2} - 1 \end{array}

重心の各座標。

\begin{array}{l} \bar{x} \\ = \frac{1}{\sqrt{2} - 1} \int_{0}^{\frac{π}{4}} \int_{\sin x}^{\cos x} x dy dx \\ = \frac{1}{\sqrt{2} - 1} \int_{0}^{\frac{π}{4}} x (\cos x - \sin x) dx \\ = \frac{1}{\sqrt{2} - 1} (\int_{0}^{\frac{π}{4}} x \cos x dx - \int_{0}^{\frac{π}{4}} x \sin x dx) \\ = \frac{1}{\sqrt{2} - 1} ({[x \sin x]}_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} \sin x dx - ({[- x \cos x]}_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{4}} \cos x dx)) \\ = \frac{1}{\sqrt{2} - 1} (\frac{π}{4} \cdot \frac{1}{\sqrt{2}} + {[\cos x]}_{0}^{\frac{π}{4}} + \frac{π}{4} \cdot \frac{1}{\sqrt{2}} - {[\sin x]}_{0}^{\frac{π}{4}}) \\ = \frac{1}{\sqrt{2} - 1} (\frac{π}{2 \sqrt{2}} + \frac{1}{\sqrt{2}} - 1 - \frac{1}{\sqrt{2}}) \\ = (\sqrt{2} + 1) (\frac{π}{2 \sqrt{2}} - 1) \\ = \frac{π}{2} + \frac{π}{2 \sqrt{2}} - \sqrt{2} - 1 \end{array}

また、

\begin{array}{l} \bar{y} \\ = \frac{1}{m} \int_{0}^{\frac{π}{4}} \int_{\sin x}^{\cos x} y dy dx \\ = \frac{1}{2 m} \int_{0}^{\frac{π}{4}} (\cos^{2} x - \sin^{2} x) dx \\ = \frac{1}{2 m} \int_{0}^{\frac{π}{4}} \cos (2 x) d x \\ = \frac{1}{4 m} {[\sin 2 x]}_{0}^{\frac{π}{4}} \\ = \frac{\sqrt{2} + 1}{4} \end{array}

コード

#!/usr/bin/env python3
from sympy import symbols, plot, sin, cos, sqrt, pi

colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']

x = symbols('x')
p = plot(sin(x), cos(x), sqrt(2) - 1, (sqrt(2) + 1) / 4,
         (x, 0, pi / 4),
         legend=True,
         show=False)

for o, color in zip(p, colors):
    o.line_color = color
p.show()
p.save('sample6.png')

入出力結果

% ./sample6.py
%