![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/circle-area.svg\")
\u9818\u57df \\(D\\) \u306e\u6761\u4ef6\u5f0f\u304b\u3089\uff0c$$D: x^2 + y^2 \\leq 1\u00a0 \\quad \\Rightarrow \\quad y^2 \\leq 1 – x^2 $$<\/p>\n
$$I = \\iint_D dx\\, dy, \\quad D: x^2 + y^2 \\leq 1$$
\n<\/p>\n
\\(\\displaystyle I = \\iint_D dx \\,dy\\) \u306f\u9818\u57df \\(D\\) \u306e\u9762\u7a4d\u3092\u8868\u3059\u306e\u3067\u3042\u3063\u305f\u3002 \\(D: x^2 + y^2 \\leq 1\\) \u3064\u307e\u308a\uff0c\u9818\u57df \\(D\\) \u306f\u534a\u5f84 \\(1\\) \u306e\u5186\u306e\u5185\u90e8\u306e\u9762\u7a4d\u3067\u3042\u308b\u304b\u3089\uff0c\u534a\u5f84 \\(r\\) \u306e\u5186\u306e\u9762\u7a4d\u306e\u516c\u5f0f \\(S = \\pi r^2\\) \u3088\u308a \\(I = \\pi\\) \u3067\u3042\u308b\u3002<\/p>\n
\u3067\u306f\u5b9f\u969b\u306b \\(\\displaystyle \\iint_{D} dx \\,dy\\) \u3092\u3069\u3046\u3084\u3063\u3066\u8a08\u7b97\u3059\u308b\u304b\uff0c2\u3064\u306e\u65b9\u6cd5\u3092\u7d39\u4ecb\u3059\u308b\u3002<\/p>\n
\u9818\u57df \\(D\\) \u306e\u6761\u4ef6\u5f0f\u304b\u3089\uff0c
$$D: x^2 + y^2 \\leq 1\u00a0 \\quad \\Rightarrow \\quad y^2 \\leq 1 – x^2 $$<\/p>\n
$$\\therefore -\\sqrt{1-x^2} \\leq y \\leq \\sqrt{1-x^2} $$<\/p>\n
\\begin{eqnarray}
I &=& \\iint_{x^2 + y^2 \\leq 1} dx \\, dy \\\\
&=& \\int_{-1}^{1}\u00a0 \\left\\{\\int_{-\\sqrt{1-x^2}}^{\\sqrt{1-x^2}} dy\\right\\} dx\\\\
&=& 2 \\int_{-1}^1 \\sqrt{1-x^2} dx
\\end{eqnarray}<\/p>\n
\u9ad8\u6821\u6570\u5b66\u3067\u306f\uff0c$y = f(x)$ \u3068 $y = g(x)$ \u304c\u533a\u9593 $a \\leq x \\leq b$ \u3067 $f(x) \\geq g(x)$ \u306e\u3068\u304d\uff0c$y = f(x), \\ y = g(x), \\ x = a, \\ x = b$ \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d $S$ \u304c<\/p>\n
$$S = \\int_a^b \\left( f(x) – g(x) \\right) \\, dx$$<\/p>\n
\u3068\u306a\u308b\u3068\u7fd2\u3063\u305f\u3068\u601d\u3046\u304c\uff0c\u3053\u306e\u5f0f\u304c\u4e0a\u8a18\u306e\u7d2f\u6b21\u7a4d\u5206\u3067\u5c0e\u304b\u308c\u305f\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n
\u3053\u3053\u3067\uff0c\\(x = \\sin\\theta\\) \u3068\u5909\u6570\u5909\u63db\u3059\u308b\u3068\uff0c\\(\\sqrt{1-x^2} = \\cos\\theta, dx = \\cos\\theta d\\theta\\)\u3002
\\begin{eqnarray}
I &=& 2 \\int_{-1}^1 \\sqrt{1-x^2} dx \\\\
\n&=& 2 \\int_{-\\pi\/2}^{\\pi\/2} \\cos^2\\theta d\\theta\\\\
&=& \\int_{-\\pi\/2}^{\\pi\/2} (1 + \\cos 2\\theta) d\\theta \\\\
&=&\\left[ \\theta + \\frac{1}{2}\\sin 2\\theta\\right]_{-\\pi\/2}^{\\pi\/2} = \\pi
\\end{eqnarray}
$$\\therefore I = \\pi$$<\/p>\n
\u305d\u3082\u305d\u3082\u5fae\u5c0f\u9762\u7a4d\u8981\u7d20 \\({\\color{green}{dx}}\\, {\\color{red}{dy}}\\) \u3068\u306f\uff0c2\u70b9 \\(P(x, y), \\ Q(x + dx, y + dy)\\) \u3092\u7d50\u3076\u7dda\u5206 $PQ$ \u3092\u5bfe\u89d2\u7dda\u3068\u3059\u308b\u5fae\u5c0f\u9577\u65b9\u5f62\u306e\u9762\u7a4d\u3067\u3042\u3063\u305f\u3002\u3053\u308c\u3092\u6975\u5ea7\u6a19\u3067\u8868\u3059\u3068\uff0c\\(P(r,\\theta), \\ Q(r + dr, \\theta + d\\theta)\\) \u3092\u3092\u7d50\u3076\u7dda\u5206 $PQ$ \u3092\u5bfe\u89d2\u7dda\u3068\u3059\u308b\u5fae\u5c0f\u9577\u65b9\u5f62\u306e\u9762\u7a4d\u3068\u306a\u308a\uff0c\\( {\\color{green}{dr}} \\times {\\color{red}{r d\\theta}}\\) \u3068\u306a\u308b\u3002<\/p>\n
![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/polar-dS.svg\")
<\/p>\n
\u6975\u5ea7\u6a19\u3067\u8868\u3059\u3068\uff0c\u9818\u57df \\(D: x^2 + y^2 \\leq 1\\) \u306f \\(D: 0 \\leq r \\leq 1, \\ 0 \\leq \\theta \\leq 2\\pi\\) \u3068\u306a\u308b\u3002\u3064\u307e\u308a\uff0c
\n\\begin{eqnarray} \\iint_{D} dx \\,dy
\n&=& \\iint_D r dr \\,d\\theta \\\\
\n&=& \\int_{0\\leq \\theta \\leq 2\\pi} \\left\\{\\int_{0\\leq r \\leq 1} r dr\\right\\} d \\theta\\\\
\n&=& \\int_0^{2 \\pi} d \\theta \\int_0^1 r dr \\\\
\n&=& 2\\pi \\times \\left[\\frac{r^2}{2}\\right]_0^1 = \\pi
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"
$$I = \\iint_D dx\\, dy, \\quad D: x^2 + y^2 \\leq 1$$<\/p>