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Transcription:

http://www.math.sci.hokudai.ac.jp/~yano/biseki2_2015/ 2015 II ( : )

f(x) : [a, b] F(x) : F (x) = f(x) ( ) F(x) F(b) F(a) f(x) b a f(x)dx = [ F(x) ] b = F(b) F(a) a f(x) x = a, x = b x S

紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

: : ( 287? 212 )

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( 3000 332 ) ( ) ナイル川 土地

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3 3 3 3 3 3 9 ( 9 ) 5 ( 3 ) + 4 ( ) = 7 ( 3 ) = 63 9 (63 1 ) 64 ( ) π πr 2 = π(9/2) 2 64 π 64 22 9 2 = 3.160493

( 1650 ) 87 48

( 1650 ) 87 48 : ( ) ( )

( 1650 ) 87 48 9 1/ 9 18 2 8 1 36 4 16 2 72 8/ 32 4 81 64 8 8(= 9 9 1 9 )

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紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

: : (1596 1650 ) (1601 1665 )

, : (1642 1727 ) (1646 1716 )

: ϵ-δ : (1789 1857 ) (1826 1866 )

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c ( ) 800 c ( ) ( ) ( )

( ) ナイル川 土地

f(x) x x = a, x = b

f(x) x x = a, x = b

f(x) x x = a, x = b

f(x) x x = a, x = b

f(x) x x = a, x = b

n [a, b] x 0 = a, x n = b a b n 1 x 1 < x 2 < < x n 1 [a, b] n : a = x 0 < x 1 < < x n 1 < x n = b = max 1 i n x i ( x i = x i x i 1 )

f(x) : [a, b] ( ) : a = x 0 < x 1 < < x n 1 < x n = b : [a, b] 1 i n [x i 1, x i ] ξ i ( ) {ξ i } n S(, {ξ i }) = f(ξ i ) x }{{} i i=1 ( x i = x i x i 1 ) ( ξ i ) f(x)

: a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 )

: a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) 12 S(, {ξ i}) = f(ξ i) x i i=1 ( x i = x i x i 1)

: a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) S(, {ξ i }) = 25 i=1 f(ξ i ) x i ( x i = x i x i 1)

: a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) S(, {ξ i }) = 49 i=1 f(ξ i ) x i ( x i = x i x i 1)

: a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) lim S(, {ξ i}) 0

f(x) : [a, b] ( ) [a, b] S(, {ξ i }) σ f(x) [a, b] ( ) b ( ) f(x)dx = σ = lim S(, {ξ i}) 0 a [a, b] f(x) 0 0

f(x) : [a, b] ( ) [a, b] S(, {ξ i }) σ f(x) [a, b] ( ) b ( ) f(x)dx = σ = lim S(, {ξ i}) 0 a [a, b] f(x) 1 2 ( )

f(x) ( ) f(x) [a, b] ξ i [x i 1, x i ] S(, {ξ i }) f(x) = { x (x 0) 1 (x = 0) [ 1, 1]

Riemann f(x) = x (0 x 1) : 0 = x 0 < x 1 < < x n 1 < x n = 1 x 0 = 0, x 1 = 1 n, x 2 = 2 n,..., x n 1 = n 1 n, x n = 1 ξ i = x i [x i 1, x i ], x i = x i x i 1 = 1 n S(, {ξ i }) n n i S(, {ξ i }) = f(ξ i ) x i = n 1 n i=1 i=1 = 1 n(n + 1) 1 n2 2 2 (n ) f(x) = sin x ξ i

( ) f(x) I F(x) f(x) ( ) a, b I b a f(x)dx = F(b) F(a) ξ i π 0 sin x dx = ( cos(π)) ( cos(0)) = 2

f(x) : [a, b] F(x) : F (x) = f(x) ( ) F(x) F(b) F(a) f(x) b a f(x)dx = [ F(x) ] b = F(b) F(a) a f(x) x = a, x = b x S

f(x) x = a, x = b x S S = b a f(x)dx b a f(x)dx = F(b) F(a) ( F (x) = f(x) )

( ) ( ) ϵ-δ ( ) ϵ-δ http://www7b.biglobe.ne.jp/~h-kuroda/pdf/text_calculus.pdf Wikimedia Commons GFDL

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