Ť

http://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%A1%E0%B8%B9%E0%B8%A5%E0%B8%90%E0%B8%B2%E0%B8%99%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%81%E0%B8%84%E0%B8%A5%E0%B8%84%E0%B8%B9%E0%B8%A5%E0%B8%B1%E0%B8%AA

Ңͧɮպ

ɮպ

f 繿ѧѹͧǧ [a, b] F 繿ѧѹѺ x [a, b]

F(x) = \int_a^x f(t)\, dt

F

Ѻء x [a, b]

f 繿ѧѹͧǧ [a, b] F 繿ѧѹ

f(x) = FѺء x [a, b]

\int_a^b f(x)\,dx = F(b) - F(a)

ŷ

f 繿ѧѹդͧǧ [a, b]. F 繿ѧѹ

f(x) = F Ѻء x [a, b]

F(x) = \int_a^x f(t)\,dt + F(a)

f(x) = \frac{d}{dx} \int_a^x f(t)\,dt

٨

ǹ 1

˹

F(x) = \int_{a}^{x} f(t) dt

x1 x1 + Δx 㹪ǧ [a, b]

F(x_1) = \int_{a}^{x_1} f(t) dt

F(x_1 + \Delta x) = \int_{a}^{x_1 + \Delta x} f(t) dt

ӷͧźѹ

F(x_1 + \Delta x) - F(x_1) = \int_{a}^{x_1 + \Delta x} f(t) dt - \int_{a}^{x_1} f(t) dt \qquad (1)

öʴ

\int_{a}^{x_1} f(t) dt + \int_{x_1}^{x_1 + \Delta x} f(t) dt = \int_{a}^{x_1 + \Delta x} f(t) dt
(鹷ͧdzԴѹ ҡѺ 鹷ͧdzͧѹ)

¢ҧ

\int_{a}^{x_1 + \Delta x} f(t) dt - \int_{a}^{x_1} f(t) dt = \int_{x_1}^{x_1 + \Delta x} f(t) dt

᷹ (1)

F(x_1 + \Delta x) - F(x_1) = \int_{x_1}^{x_1 + \Delta x} f(t) dt \qquad (2)

ɮպѺԹõ c 㹪ǧ [x1, x1 + Δx]

\int_{x_1}^{x_1 + \Delta x} f(t) dt = f(c) \Delta x

᷹ŧ (2)

F(x_1 + \Delta x) - F(x_1) = f(c) \Delta x \,

÷ͧҧ Δx

\frac{F(x_1 + \Delta x) - F(x_1)}{\Delta x} = f(c)
ѧࡵâҧ ѵǹԧŵҧͧǵѹ (Newton's difference quotient) ͧ F x1

Ե Δx → 0 ͧҧͧ

\lim_{\Delta x \to 0} \frac{F(x_1 + \Delta x) - F(x_1)}{\Delta x} = \lim_{\Delta x \to 0} f(c)

âҧ¨͹ؾѹͧ F x1

F

Եͧâҧ Ҩɮպ squeeze c 㹪ǧ [x1, x1 + Δx] ѧ x1cx1 + Δx

ҡ \lim_{\Delta x \to 0} x_1 = x_1 \lim_{\Delta x \to 0} x_1 + \Delta x = x_1

ɮպ squeeze

\lim_{\Delta x \to 0} c = x_1

᷹ŧ (3)

F

ѧѹ f դͧ c ѧ öԵ᷹㹿ѧѹ ѧ

F

þ٨

(Leithold et al, 1996)

ǹ 2

仹ͺ٨Ե ͧѹ-Һٵ

ҾʴǤԴͧ ѹ-Һٵ 㹡ûҳ鹷ҿ ¡ҿ觨ӹǹҡ

f 繿ѧѹդͧǧ [a, b] F 繻ҹؾѹͧ f ԨóҹԾ仹

F(b) - F(a)\,

a = x_0 < x_1 < x_2 < \ldots < x_{n-1} < x_n = b

F(b) - F(a) = F(x_n) - F(x_0) \,

Ǻǡź¨ӹǹǡѹ

\begin{matrix} F(b) - F(a) & = & F(x_n)\,+\,[-F(x_{n-1})\,+\,F(x_{n-1})]\,+\,\ldots\,+\,[-F(x_1) + F(x_1)]\,-\,F(x_0) \, \\
& = & [F(x_n)\,-\,F(x_{n-1})]\,+\,[F(x_{n-1})\,+\,\ldots\,-\,F(x_1)]\,+\,[F(x_1)\,-\,F(x_0)] \, \end{matrix}

¹

F(b) - F(a) = \sum_{i=1}^n [F(x_i) - F(x_{i-1})] \qquad (1)

Ҩɮպ

f 繿ѧѹդͧǧ [a, b] ͹ؾѹ캹ǧ (a, b) c (a, b)

f

Ш

f

ѧѹ F 繿ѧѹ͹ؾѹ㹪ǧ [a, b] ѧ ѹ͹ؾѹդͧЪǧ xi-1 ɮպ

F(x_i) - F(x_{i-1}) = F

᷹ŧ (1)

F(b) - F(a) = \sum_{i=1}^n [F

ҡ F xixi − 1 ö¹ٻ Δx ͧ觡 i

F(b) - F(a) = \sum_{i=1}^n [f(c_i)(\Delta x_i)] \qquad (2)

ѧࡵҡѧ͸Ժ¾鹷ͧ׹ դҧٳ٧ ҡǡ鹷ҹҴ¡ѹ ҡɮպ ׹ٻ͸Ժ¤һҳͧǹͧ ѧࡵա Δxi 繵ͧ͹ѹ㹷ءҢͧ i ¤Ҥҧͧ繵ͧҡѹ 觷ҵͧӤͻҳ駴¨ӹǹ n ٻ ͢Ҵͧǹҧŧ n դҡ Դǹҧҡ ͤͺ鹷 Ҩ鹷ԧͧ

¡ԵͧԾͤ¢ͧǹҧ ٹ Ҩ ԾѹẺѹ 蹤 Ե͢Ҵǹ˭شٹ ǹբҴŧ Шӹǹǹ͹ѹ

ѧ ҨԵ价ͧҧͧ (2)

\lim_{\| \Delta \| \to 0} F(b) - F(a) = \lim_{\| \Delta \| \to 0} \sum_{i=1}^n [f(c_i)(\Delta x_i)]\,dx

F(b) F(a) ҧ鹡Ѻ ||Δ|| ѧ Եͧҧ¨֧ҡѺ F(b) - F(a)

F(b) - F(a) = \lim_{\| \Delta \| \to 0} \sum_{i=1}^n [f(c_i)(\Delta x_i)]

йԾҧҢͧ ¶֧ԹԡŢͧ f ҡ a b ѧ Ҩ

F(b) - F(a) = \int_{a}^{b} f(x)\,dx

þ٨

ҧ

ҧ Ҥسͧäӹdz

\int_2^5 x^2\;\mathrm{d}x

f(x) = x2 Ҩ F(x)=\frac{x^3}{3} 繻ҹؾѹ ѧ

\int_2^5 x^2\;\mathrm{d}x = F(5) - F(2) = {125 \over 3} - {8 \over 3} = {117 \over 3} = 39

ҵͧ

\int_1^3 \frac{dx}{x}=\big[\ln|x|\big]_1^3 =\ln 3-\ln1=\ln 3

·

繵ͧ f ͧʹ駪ǧ ѧǹ 1 ͧɮպС f 繿ѧѹöԾѹມǧ [a,b] x0 繨ӹǹ㹪ǧ [a,b] f ͧ x0

F(x) = \int_a^x f(t)\;\mathrm{d}t

ö͹ؾѹѺ x = x0 F(x0) = f(x0) ö͹䢢ͧ f §öһԾѹ㹵˹觹 㹡óչ öػҿѧѹ F ö͹ؾѹͺء F'(x) = f(x) ͺء ҧ¡ɮչ ɮպ͹ؾѹͧມ

ǹ 2ͧɮպ繨ԧѺءѧѹ f öһԾѹມ ջҹؾѹ F (ءѧѹ͹ؾѹ)

ǹͧɮպͧ觡Ƕ֧ԴͼԴҴ繻Ծѹöͧ繹·仢ͧɮպŰҹͧŤ

շɮպ˹Ѻѧѹԧ͹: U ૵Դ \mathbb{C} f:U\to\mathbb{C} 繿ѧѹ Ծѹ F U ѧѺ \gamma: [a,b] \to U Ծѹ駨Фӹdzҡ

\oint_{\gamma} f(z) \;\mathrm{d}z = F(\gamma(b)) - F(\gamma(a))

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