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·Äɮպ·¹ÕéÇèÒäÇéÇèÒ

ãËé f à»ç¹¿Ñ§¡ìªÑ¹µèÍà¹×èͧº¹ªèǧ [a, b] ¶éÒ F à»ç¹¿Ñ§¡ìªÑ¹·Õè¹ÔÂÒÁÊÓËÃѺ x ·ÕèÍÂÙèã¹ [a, b] ÇèÒ

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

áÅéÇ

F

ÊÓËÃѺ·Ø¡ x ·ÕèÍÂÙèã¹ [a, b]

ãËé f à»ç¹¿Ñ§¡ìªÑ¹µèÍà¹×èͧº¹ªèǧ [a, b] ¶éÒ F à»ç¹¿Ñ§¡ìªÑ¹·Õè

f(x) = FÊÓËÃѺ·Ø¡ x ·ÕèÍÂÙèã¹ [a, b]

áÅéÇ

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

¼Å·ÕèµÒÁÁÒ

ãËé f à»ç¹¿Ñ§¡ìªÑ¹·ÕèÁÕ¤ÇÒÁµèÍà¹×èͧº¹ªèǧ [a, b]. ¶éÒ F à»ç¹¿Ñ§¡ìªÑ¹·Õè

f(x) = F ÊÓËÃѺ·Ø¡ x ·ÕèÍÂÙèã¹ [a, b]

áÅéÇ

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

áÅÐ

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

º·¾ÔÊÙ¨¹ì

Êèǹ·Õè 1

¡Ó˹´ãËé

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

ãËé x1 áÅÐ x1 + Δx ÍÂÙè㹪èǧ [a, b] ¨Ðä´é

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

áÅÐ

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

¹Ó·Ñé§ÊͧÊÁ¡ÒÃÁÒź¡Ñ¹ä´é

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)

àÃÒÊÒÁÒöáÊ´§ä´éÇèÒ

\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
(¼ÅÃÇÁ¾×é¹·Õè¢Í§ºÃÔàdz·ÕèÍÂÙèµÔ´¡Ñ¹ ¨Ðà·èҡѺ ¾×é¹·Õè¢Í§ºÃÔàdz·Ñé§ÊͧÃÇÁ¡Ñ¹)

ÂéÒ¢éÒ§ÊÁ¡ÒÃä´é

\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

¹Óä»á·¹¤èÒã¹ (1) ¨Ðä´é

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

µÒÁ·Äɮպ·¤èÒà©ÅÕèÂÊÓËÃѺ¡ÒÃÍÔ¹·Ôà¡Ãµ ¨ÐÁÕ c ÍÂÙè㹪èǧ [x1, x1 + Δx] ·Õè·ÓãËé

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

á·¹¤èÒŧ㹠(2) ä´é

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

ËÒ÷Ñé§Êͧ¢éÒ§´éÇ Δx ¨Ðä´é

\frac{F(x_1 + \Delta x) - F(x_1)}{\Delta x} = f(c)
ÊѧࡵÇèÒÊÁ¡ÒâéÒ§«éÒ ¤×Í ÍѵÃÒÊèǹàªÔ§¼ÅµèÒ§¢Í§¹Ôǵѹ (Newton's difference quotient) ¢Í§ F ·Õè x1

ãÊèÅÔÁÔµ Δx → 0 ·Ñé§Êͧ¢éÒ§¢Í§ÊÁ¡ÒÃ

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

ÊÁ¡ÒâéÒ§«éÒ¨Ðà»ç¹Í¹Ø¾Ñ¹¸ì¢Í§ F ·Õè x1

F

à¾×èÍËÒÅÔÁÔµ¢Í§ÊÁ¡ÒâéÒ§¢ÇÒ àÃÒ¨Ðãªé·Äɮպ· squeeze à¾ÃÒÐÇèÒ c ÍÂÙè㹪èǧ [x1, x1 + Δx] ´Ñ§¹Ñé¹ x1cx1 + Δx

¨Ò¡ \lim_{\Delta x \to 0} x_1 = x_1 áÅÐ \lim_{\Delta x \to 0} x_1 + \Delta x = x_1

µÒÁ·Äɮպ· squeeze ¨Ðä´éÇèÒ

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

á·¹¤èÒŧ㹠(3) ¨Ðä´é

F

¿Ñ§¡ìªÑ¹ f ÁÕ¤ÇÒÁµèÍà¹×èͧ·Õè c ´Ñ§¹Ñé¹ àÃÒÊÒÁÒö¹ÓÅÔÁԵ᷹㹿ѧ¡ìªÑ¹ä´é ´Ñ§¹Ñé¹

F

¨º¡ÒþÔÊÙ¨¹ì

(Leithold et al, 1996)

Êèǹ·Õè 2

µèÍ仹Õé¤×ͺ·¾ÔÊÙ¨¹ìÅÔÁÔµâ´Â ¼ÅÃÇÁ¢Í§ÃÕÁѹ¹ì-´ÒºÙµì

ÿ
ÀÒ¾áÊ´§á¹Ç¤Ô´¢Í§ ¼ÅÃÇÁÃÕÁѹ¹ì-´ÒºÙµì «Öè§ãªé㹡ÒûÃÐÁÒ³¾×é¹·ÕèÀÒÂãµé¡ÃÒ¿ã´ æ ´éÇ¡ÃÒ¿á·è§¨Ó¹Ç¹ÁÒ¡

ãËé f à»ç¹¿Ñ§¡ìªÑ¹·ÕèÁÕ¤ÇÒÁµèÍà¹×èͧº¹ªèǧ [a, b] áÅÐ F à»ç¹»¯ÔÂҹؾѹ¸ì¢Í§ f ¾Ô¨ÒóҹԾ¨¹ìµèÍ仹Õé

F(b) - F(a)\,

ãËé a = x_0 < x_1 < x_2 < \ldots < x_{n-1} < x_n = b ¨Ðä´é

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

áÅéǺǡáÅÐź´éǨӹǹà´ÕÂǡѹ ¨Ðä´é

\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}

à¢Õ¹ãËÁèà»ç¹

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

àÃÒ¨Ðãªé·Äɮպ·¤èÒà©ÅÕè «Öè§¡ÅèÒÇÇèÒ

ãËé f à»ç¹¿Ñ§¡ìªÑ¹·ÕèÁÕ¤ÇÒÁµèÍà¹×èͧº¹ªèǧ [a, b] áÅÐÁÕ͹ؾѹ¸ìº¹ªèǧ (a, b) áÅéÇ ¨ÐÁÕ c ÍÂÙèã¹ (a, b) ·Õè·ÓãËé

f

áÅШÐä´é

f

¿Ñ§¡ìªÑ¹ F à»ç¹¿Ñ§¡ìªÑ¹·ÕèËÒ͹ؾѹ¸ìä´é㹪èǧ [a, b] ´Ñ§¹Ñé¹ Áѹ¨ÐËÒ͹ؾѹ¸ìáÅÐÁÕ¤ÇÒÁµèÍà¹×èͧº¹áµèÅЪèǧ xi-1 ä´é µÒÁ·Äɮպ·¤èÒà©ÅÕè ¨Ðä´é

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

á·¹¤èÒŧ㹠(1) ¨Ðä´é

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

¨Ò¡ F áÅÐ xixi − 1 ÊÒÁÒöà¢Õ¹ã¹ÃÙ» Δx ¢Í§¼Åáºè§¡Ñé¹ i

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

ÊѧࡵÇèÒàÃÒ¡ÓÅѧ͸ԺÒ¾×é¹·Õè¢Í§ÊÕèàËÅÕèÂÁ¼×¹¼éÒ â´ÂÁÕ¤ÇÒÁ¡ÇéÒ§¤Ù³¤ÇÒÁÊÙ§ áÅÐàÃÒ¡çºÇ¡¾×é¹·ÕèàËÅèÒ¹Ñé¹à¢éÒ´éÇ¡ѹ ¨Ò¡·Äɮպ·¤èÒà©ÅÕè ÊÕèàËÅÕèÂÁ¼×¹¼éÒáµèÅÐÃٻ͸ԺÒ¤èÒ»ÃÐÁÒ³¢Í§Êèǹ¢Í§àÊé¹â¤é§ ÊѧࡵÍÕ¡ÇèÒ Δxi äÁè¨Óà»ç¹µéͧàËÁ×͹¡Ñ¹ã¹·Ø¡æ¤èҢͧ i ËÃ×ÍËÁÒ¤ÇÒÁÇèÒ¤ÇÒÁ¡ÇéÒ§¢Í§ÊÕèàËÅÕèÂÁ¹Ñé¹äÁè¨Óà»ç¹µéͧà·èҡѹ ÊÔè§·ÕèàÃÒµéͧ·Ó¤×Í»ÃÐÁÒ³àÊé¹â¤é§´éǨӹǹÊÕèàËÅÕèÂÁ n ÃÙ» àÁ×èÍ¢¹Ò´¢Í§ÊèǹµèÒ§æàÅç¡Å§ áÅÐ n ÁÕ¤èÒÁÒ¡¢Öé¹ ·ÓãËéà¡Ô´ÊèǹµèÒ§æÁÒ¡¢Öé¹ à¾×èͤÃͺ¤ÅØÁ¾×é¹·Õè àÃÒ¨ÐÂÔè§à¢éÒã¡Åé¾×é¹·Õè¨ÃÔ§æ¢Í§àÊé¹â¤é§

â´Â¡ÒÃËÒÅÔÁÔµ¢Í§¹Ô¾¨¹ì¹Õéà»ç¹àÁ×èͤèÒà©ÅÕè¢ͧÊèǹµèÒ§æ¹Õé à¢éÒã¡ÅéÈÙ¹Âì àÃÒ¨Ðä´é »ÃԾѹ¸ìẺÃÕÁѹ¹ì ¹Ñ蹤×Í àÃÒËÒÅÔÁÔµàÁ×èÍ¢¹Ò´Êèǹ·ÕèãË­è·ÕèÊØ´à¢éÒã¡ÅéÈÙ¹Âì ¨Ðä´éÊèǹÍ×è¹æÁÕ¢¹Ò´àÅç¡Å§ áÅШӹǹÊèǹà¢éÒã¡Åé͹ѹµì

´Ñ§¹Ñé¹ àÃÒ¨ÐãÊèÅÔÁԵ价Ñé§Êͧ¢éÒ§¢Í§ÊÁ¡Òà (2) ¨Ðä´é

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

·Ñé§ F(b) áÅÐ F(a) µèÒ§¡çäÁè¢Ö鹡Ѻ ||Δ|| ´Ñ§¹Ñé¹ ÅÔÁÔµ¢Í§¢éÒ§«éÒ¨֧à·èҡѺ F(b) - F(a)

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

áÅйԾ¨¹ì·Ò§¢ÇҢͧÊÁ¡Òà ËÁÒ¶֧ÍÔ¹·Ô¡ÃÑŢͧ f ¨Ò¡ a ä» b ´Ñ§¹Ñé¹ àÃÒ¨Ðä´é

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

¨º¡ÒþÔÊÙ¨¹ì

µÑÇÍÂèÒ§

µÑÇÍÂèÒ§àªè¹ ¶éҤسµéͧ¡ÒäӹdzËÒ

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

ãËé f(x) = x2 àÃÒ¨Ðä´é F(x)=\frac{x^3}{3} à»ç¹»¯ÔÂҹؾѹ¸ì ´Ñ§¹Ñé¹

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

¶éÒàÃÒµéͧ¡ÒÃËÒ

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

¹Ñ·ÑèÇä»

àÃÒäÁè¨Óà»ç¹µéͧãËé f µèÍà¹×èͧµÅÍ´·Ñ駪èǧ ´Ñ§¹Ñé¹Êèǹ·Õè 1 ¢Í§·Äɮպ·¨Ð¡ÅèÒÇÇèÒ ¶éÒ f à»ç¹¿Ñ§¡ìªÑ¹·ÕèÊÒÁÒöËÒ»ÃԾѹ¸ìàÅÍມº¹ªèǧ [a,b] áÅÐ x0 à»ç¹¨Ó¹Ç¹ã¹ªèǧ [a,b] «Öè§ f µèÍà¹×èͧ·Õè x0 ¨Ðä´é

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

ÊÒÁÒöËÒ͹ؾѹ¸ìä´éÊÓËÃѺ x = x0 áÅÐ F(x0) = f(x0) àÃÒÊÒÁÒö¤ÅÒÂà§×è͹䢢ͧ f à¾Õ§á¤èãËéÊÒÁÒöËÒ»ÃԾѹ¸ìä´éã¹µÓá˹觹Ñé¹ ã¹¡Ã³Õ¹Ñé¹ àÃÒÊÒÁÒöÊÃØ»ä´éÇèҿѧ¡ìªÑ¹ F ¹Ñè¹ÊÒÁÒöËÒ͹ؾѹ¸ìä´éà¡×ͺ·Ø¡·Õè áÅÐ F'(x) = f(x) ¨Ðà¡×ͺ·Ø¡·Õè ºÒ§·ÕàÃÒàÃÕ¡·ÄɮչÕéÇèÒ ·Äɮպ·Í¹Ø¾Ñ¹¸ì¢Í§àÅÍມ

Êèǹ·Õè 2¢Í§·Äɮպ·¹Õéà»ç¹¨ÃÔ§ÊÓËÃѺ·Ø¡¿Ñ§¡ìªÑ¹ f ·ÕèÊÒÁÒöËÒ»ÃԾѹ¸ìàÅÍມä´é áÅÐÁÕ»¯ÔÂҹؾѹ¸ì F (äÁèãªè·Ø¡¿Ñ§¡ìªÑ¹·ÕèËÒ͹ؾѹ¸ìä´é)

Êèǹ¢Í§·Äɮպ·¢Í§à·ÂìàÅÍÃì«Öè§¡ÅèÒǶ֧¾¨¹ì·Õèà¡Ô´¢éͼԴ¾ÅÒ´à»ç¹»ÃԾѹ¸ìÊÒÁÒöÁͧä´éà»ç¹¹Ñ·ÑèÇ仢ͧ·Äɮպ·ÁÙŰҹ¢Í§á¤Å¤ÙÅÑÊ

ÁÕ·Äɮպ·Ë¹Öè§ÊÓËÃѺ¿Ñ§¡ìªÑ¹àªÔ§«é͹: ãËé U à»ç¹à«µà»Ô´ã¹ \mathbb{C} áÅÐ f:U\to\mathbb{C} à»ç¹¿Ñ§¡ìªÑ¹·ÕèÁÕ »ÃԾѹ¸ìâÎâÅÁÍÃì¿ F ã¹ U ´Ñ§¹Ñé¹ÊÓËÃѺàÊé¹â¤é§ \gammaÿ: [a,b] \to U »ÃԾѹ¸ìàÊé¹â¤é§¨Ð¤Ó¹Ç³ä´é¨Ò¡

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

·Äɮպ·ÁÙŰҹ¢Í§á¤Å¤ÙÅÑÊÊÒÁÒöÇÒ§¹Ñ·ÑèÇä»ãËé¡Ñº »ÃԾѹ¸ìàÊé¹â¤é§áÅо×é¹¼ÔÇã¹ÁÔµÔ·ÕèÊÙ§¡ÇèÒáÅк¹áÁ¹Ôâ¿Å´ìä´é

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