$$ \\int_a^b f(x) \\, dx = \\Bigl[ F(x) \\Bigr]^b_a \\equiv F(b)\u00a0 -F(a)$$
\u307e\u305f\uff0c
$$\\int_a^b f(x) \\, dx = \\int_a^b f(t) \\, dt = \\int_a^b f(*) \\, d*$$\u306e\u3088\u3046\u306b \\(*\\) \u306e\u90e8\u5206\u306e\u5909\u6570\u306e\u3053\u3068\u3092\u300c\u7a4d\u5206\u5909\u6570\u300d\u3068\u3044\u3046\u304c\uff0c\u3053\u306e\u7a4d\u5206\u5909\u6570\u306b\u306f\u3069\u3093\u306a\u5909\u6570\u3092\u4f7f\u3063\u3066\u3082\u3088\u3044\u3002<\/p>\n
\u5b9a\u7a4d\u5206\u306e\u6027\u8cea<\/h3>\n
\u5b9a\u7a4d\u5206\u306f\uff0c\u7a4d\u5206\u533a\u9593\u306b\u95a2\u3057\u3066\u52a0\u6cd5\u6027\u3092\u6709\u3059\u308b\uff1a\u3064\u307e\u308a\uff0c
$$\\int_a^{\\color{red}b} f(x)\\, dx + \\int_{\\color{red}b}^c f(x)\\, dx = \\int_a^c f(x)\\,dx$$<\/p>\n
\u307e\u305f\uff0c
$$\\int_a^a f(x)\\, dx = 0$$<\/p>\n
\u4e0a\u8a18\u306e2\u3064\u306e\u6027\u8cea\u304b\u3089\u305f\u3060\u3061\u306b\uff0c\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u308f\u304b\u308b\u3002
$$\\int_a^b f(x)\\, dx\u00a0 = -\\int_b^a f(x)\\,dx$$<\/p>\n
\u3055\u3089\u306b\uff0c\u4e0d\u5b9a\u7a4d\u5206\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u3057\u305f\u7a4d\u5206\u306e\u7dda\u5f62\u6027\u306f\uff0c\u5b9a\u7a4d\u5206\u306b\u5bfe\u3057\u3066\u3082\u305d\u306e\u307e\u307e\u6210\u308a\u7acb\u3064\u3002\\(k\\) \u3092\u5b9a\u6570\u3068\u3059\u308b\u3068\uff0c
$$ \\int_a^b k f(x)\\, dx = k \\int_a^b f(x)\\, dx$$
$$\\int_a^b \\left\\{f(x) + g(x)\\right\\}\\, dx = \\int_a^b f(x)\\, dx + \\int_a^b g(x)\\, dx$$<\/p>\n
\u5b9a\u7a4d\u5206\u304c\u9762\u7a4d\u3092\u4e0e\u3048\u308b\u3053\u3068<\/h3>\n
![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/Sekibun-Area.svg\")
<\/p>\n
\u4e0a\u56f3\u306e\u3088\u3046\u306b\uff0c\\(y = f(x) > 0\\)\uff0c\\( y = 0\\)\uff08\\(x\\) \u8ef8\uff09\uff0c\\(x = a\\)\uff0c\u304a\u3088\u3073 \\(x = x\\) \u3067\u56f2\u307e\u308c\u305f\u9ec4\u8272\u3044\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S(x)\\) \u3068\u3059\u308b\u3002<\/p>\n
\u4eca\uff0c\\(x \\rightarrow x + \\Delta x\\) \u3068\u5909\u5316\u3057\u305f\u3068\u304d\u306e\u9762\u7a4d\u306e\u5909\u5316\u5206\uff08\u4e0a\u56f3\u306e\u7070\u8272\u90e8\u5206\uff09\u3092
\n$$\\Delta S \\equiv S(x + \\Delta x) -S(x)$$ \u3068\u3059\u308b\u3002\u4eca\uff0c\u7740\u76ee\u3057\u3066\u3044\u308b\u7bc4\u56f2\u3067 \\(f'(x) \\geq 0\\) \u3068\u3059\u308b\u3068\uff0c\u56f3\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c
\n$$ {\\color{blue}{f(x) \\Delta x}} \\leq \\Delta S \\leq {\\color{red}{f(x+ \\Delta x) \\Delta x}}$$<\/p>\n
\u5404\u9805\u3092 \\(\\Delta x\\) \u3067\u5272\u3063\u3066 \\(\\Delta x \\rightarrow 0\\) \u306e\u6975\u9650\u3092\u3068\u308b\u3068\uff0c
\n$$ f(x) \\leq \\lim_{\\Delta x \\rightarrow 0} \\frac{\\Delta S}{\\Delta x} = \\frac{dS}{dx} \\leq \\lim_{\\Delta x \\rightarrow 0}f(x + \\Delta x) = f(x)$$
\n$$\\therefore\\ \\ \\frac{dS}{dx} = f(x)$$<\/p>\n
\u3064\u307e\u308a\uff0c\\(S(x)\\) \u306f \\(f(x)\\) \u306e\u539f\u59cb\u95a2\u6570\uff08\u306e\u3072\u3068\u3064\uff09\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\uff0c
\n$$S(x) = \\int f(x)\\, dx = F(x) + C$$\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\(x = a\\) \u306e\u3068\u304d\uff0c\u9762\u7a4d\u306f\u30bc\u30ed\u3067\u3042\u308b\u306e\u3067
\n$$S(a) = F(a) + C = 0, \\quad \\therefore\\ \\ C = -F(a), \\quad S(x) = F(x) -F(a)$$<\/p>\n
\u5b9a\u7a4d\u5206\u306e\u8a18\u6cd5\u3092\u601d\u3044\u51fa\u3059\u3068\uff0c
\n$$ S(x) = F(x) -F(a) = \\Bigl[F(x)\\Bigr]_a^x = \\int_a^x f(t)\\,dt$$<\/p>\n
\u3053\u306e\u7d50\u679c\u304b\u3089\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
1. \u5b9a\u7a4d\u5206 \\(\\displaystyle\u00a0 \\int_a^b f(x)\\,dx\\) \u306f\uff0c\\(y=f(x)\\ (>0), \\ \\ y = 0, \\ \\ x = a, \\ \\ x = b \\ (>a)\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u8868\u3059\u3053\u3068\u3002<\/p>\n
2. \u4e0d\u5b9a\u7a4d\u5206 \\(\\displaystyle\u00a0 \\int f(x)\\,dx\\) \u306f\uff0c\u5b9a\u7a4d\u5206\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u3002\u7a4d\u5206\u306e\u4e0b\u7aef \\(a\\) \u306f\u4efb\u610f\u306e\u5b9a\u6570\u306a\u306e\u3067\uff0c\u7701\u7565\u3057\u3066\u66f8\u304f\u5834\u5408\u3082\u3042\u308b\u3002\u7a4d\u5206\u306e\u4e0b\u7aef\u304c\u4e0d\u5b9a<\/span>\u306a\u306e\u3067\u4e0d\u5b9a\u7a4d\u5206<\/span><\/strong><\/span>\u3002 \u6ce8\u610f\uff1a\\(\\displaystyle \\int^x f(x)\\,dx\\) \u306a\u3069\u3068\u66f8\u304f\u3068\u7a4d\u5206\u306e\u4e0a\u7aef\u306e\u5024\u3067\u3042\u308b\\(x\\) \u304c\u3042\u305f\u304b\u3082\u7a4d\u5206\u5909\u6570\u3068\u6df7\u540c\u3055\u308c\u308b\u306e\u3067\uff0c\\(\\displaystyle \\int^x f(t)\\,dt\\) \u306a\u3069\u3068\u66f8\u304f\u306e\u304c\u5409\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u5b9a\u7a4d\u5206\u306e\u5b9a\u7fa9 <\/p>\n \u533a\u9593 \\([a, b]\\) \u306b\u304a\u3051\u308b \\(f(x)\\) \u306e\u5b9a\u7a4d\u5206\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3002$$ \\int_a^b f(x) \\, dx = \\Bigl[ F(x) \\Bigr]^b_a \\equiv F(b)\u00a0 -F(a)$$\u307e\u305f\uff0c$$\\int_a^b f(x) \\, dx = \\int_a^b f(t) \\, dt = \\int_a^b f(*) \\, d*$$\u306e\u3088\u3046\u306b \\(*\\) \u306e\u90e8\u5206\u306e\u5909\u6570\u306e\u3053\u3068\u3092\u300c\u7a4d\u5206\u5909\u6570\u300d\u3068\u3044\u3046\u304c\uff0c\u3053\u306e\u7a4d\u5206\u5909\u6570\u306b\u306f\u3069\u3093\u306a\u5909\u6570\u3092\u4f7f\u3063\u3066\u3082\u3088\u3044\u3002<\/p>
\n$$\\int f(x)\\,dx = \\int_a^{x} f(t)\\,dt = \\int^x f(t)\\,dt$$<\/p>\n