CẢI CÁCH QUỐC GIA

CẢI CÁCH QUỐC GIA

Pétítísốềếsố Đ í é t ế tí é ũ t ề ệ tí ì t tớ ệ tí ị ột ớ t ề tể tí ì trụ sẽ t tớ ệ ớ tí ớ u = f(x, y) ị tụ tr ề D ó ị tr Oxy t í V ớ ở ớ D (Oxy) tr u = f(x, y) t q ó ờ s s s ớ Oz tự tr D ì ẽ í t ủ tí é số u = f(x, y) ị tr ề D ó ị D ột tỳ ý t n ỏ 1 , 2 , ., n r ỗ S i ột ể tỳ ý M i (x i , y i ) ổ I n = n i=1 f(x i , y i )S( i ) ợ ọ tổ tí ủ số f(x, y) tr ề D ọ d i = ờ í ủ i sup d(M, M ), M, M i ế n s max{d i } 0 I n tớ ột ớ ữ I ụ tộ ề D ể M i tr ỗ i tì ớ ợ ọ tí é ủ số f(x, y) tr ề D ý ệ D f(x, y)dS := lim max{d i }0 I n D ề tí f ớ tí D f(x, y)dS tồ t t ó f(x, y) tí tr ề D ì tí é ụ tộ ề D t ỏ t ó tể D ở ọ ờ t s s ớ trụ t ộ ó dS = dxdy ó tể ết D f(x, y)dS = D f(x, y)dxdy ú ý ế f(x, y) tụ tr ề ó ị D tì tí tr ề í t ớ tết ó t tr tí ề tí tí é ũ ó ột số tí t ố tí ị ế tí D (kf + lg)dxdy = k D fdxdy + l D gdxdy ộ tí ế ề D = D 1 D 2 tr int(D 1 )int(D 2 ) = tì D fdxdy = D 1 fdxdy + D 2 fdxdy. tứ tự f g,(x, y) D t❤×  D fdxdy ≤  D gdxdy ✹✳ ◆Õ✉ m ≤ f(x, y) ≤ M,∀(x, y) ∈ D✱ m ✈➭ M ❧➭ ❤➺♥❣ sè✱ t❤× mS(D) ≤  D fdxdy ≤ M S(D) ✺✳ ➜▲ ✈Ò ❣✐➳ trÞ tr✉♥❣ ❜×♥❤ ◆Õ✉ f(x, y) ❧✐➟♥ tô❝ tr♦♥❣ ♠✐Ò♥ ➤ã♥❣✱ ❜Þ ❝❤➷♥ D t❤× ∃M 0 (x 0 , y 0 ) ∈ D : f(M 0 ) =  D fdxdy S(D) . ✶✳✸✳ ❈➳❝❤ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❦Ð♣ tr♦♥❣ t♦➵ ➤é ✈✉➠♥❣ ❣ã❝ ❚r♦♥❣ tÝ❝❤ ♣❤➞♥ ♠ét ❧í♣✱ t❛ ➤➲ ➤Ò ❝❐♣ ➤Õ♥ ❜➭✐ t♦➳♥ tÝ♥❤ t❤Ó tÝ❝❤ ✈❐t t❤Ó t❤❡♦ t✐Õt ❞✐Ö♥ ♥❣❛♥❣ ❝ñ❛ ♥ã✳ t✐Õ♣ tô❝ ❣✐➯✐ t❤Ý❝❤ ❜➺♥❣ ❤×♥❤ ❤ä❝ tÝ❝❤ ♣❤➞♥ ✷ ❧í♣ ♥❤➢ ❧➭ t❤Ó tÝ❝❤ ❤×♥❤ trô ❝♦♥❣✱ t❛ sÏ ➤➢❛ ✈✐Ö❝ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ✷ ❧í♣ ✈Ò tÝ❝❤ ♣❤➞♥ ❧➷♣✳ • ▼✐Ò♥ D ❧➭ ❤×♥❤ ❝❤÷ ♥❤❐t  a ≤ x ≤ b c ≤ y ≤ d I = b  a   d  c f(x, y)dy   dx := b  a dx d  c f(x, y)dy ❍♦➷❝ I = d  c   b  a f(x, y)dx   dy := d  c dy b  a f(x, y)dx ❦❤✐ tÝ♥❤ tÝ❝❤ ♣❤➞♥ tr♦♥❣ ♥❣♦➷❝✱ t❤❡♦ ❜✐Õ♥ ♥➭② t❤× ❝♦✐ ❜✐Õ♥ ❦✐❛ ❧➭ ❤➺♥❣ sè✳ • ▼✐Ò♥ D ❧➭ ♠✐Ò♥ ①➳❝ ➤Þ♥❤ ❜ë✐ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝      a ≤ x ≤ b y 1 (x) ≤ y ≤ y 2 (x) y 1 (x), y 2 (x) ∈ C[a, b]  a ≤ x ≤ b c ≤ y ≤ d I = b  a    y 2 (x)  y 1 (x) f(x, y)dy    dx := b  a dx y 2 (x)  y 1 (x) f(x, y)dy ✸ • ▼✐Ò♥ D ❧➭ ♠✐Ò♥ ①➳❝ ➤Þ♥❤ ❜ë✐ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝      c ≤ y ≤ d x 1 (y) ≤ x ≤ x 2 (y) x 1 (y), x 2 (y) ∈ C[c, d] I = d  c    x 2 (y)  x 1 (y) f(x, y)dx    dy := d  c dx y 2 (x)  y 1 (x) f(x, y)dy • ▼✐Ò♥ D ❧➭ ♠✐Ò♥ ♣❤➻♥❣ ❜✃t ❦ú ❈❤✐❛ D t❤➭♥❤ ❝➳❝ ♠✐Ò♥ ♥❤á rê✐ ♥❤❛✉✱ ❝ã ❞➵♥❣ ë tr➟♥✳ ❱Ý ❞ô✿ ❇➭✐ t❐♣ ✻✳✶✱ ✻✳✷✱ ✻✳✸✳ ✶✳✹✳ ➜æ✐ ❜✐Õ♥ tr♦♥❣ tÝ❝❤ ♣❤➞♥ ❦Ð♣ ✶✳✹✳✶✳ ❈➠♥❣ t❤ø❝ tæ♥❣ q✉➳t ❚❛ ❞ï♥❣ ♣❤Ð♣ ➤æ✐ ❜✐Õ♥ s❛✉✿  x = x(u, v) y = y(u, v) (1) t❤♦➯ ♠➲♥            ∃x(u, v), y(u, v), x  u , x  v , y  u , y  v ∈ C(D  ) (1) ①➳❝ ➤Þ♥❤ ♠ét s♦♥❣ ➳♥❤ tõ D ❧➟♥ D  ✳ ➜Þ♥❤ t❤ø❝ ❏❛❝♦❜✐ ✭❏❛❝♦❜✐❛♥✮ J = D(x,y) D(u,v) =      x  u x  v y  u y  v      = 0 tr♦♥❣ D  . ❑❤✐ ➤ã I =  D f(x, y)dxdy =  D  f[x(u, v), y(u, v)] |J|dudv ❍×♥❤ ✈Ï ●✐➯✐ t❤Ý❝❤ ❝ñ❛ ↕str➠❣r❛t①❦✐ ✹ ❱í✐  x = x(u, v) y = y(u, v) (1) A 1 (u, v) A 2 (u + du, v) A 3 (u + du, v + dv) A 4 (u, v + dv) B 1 [x(u, v), y(u, v)], B 2 [x(u + du, v), y(u + du, v)] B 3 [x(u + du, v + dv), y(u + du, v + dv)] B 4 [x(u, v + dv), y(u, v + dv)] ◆Õ✉ ❣✐í✐ ❤➵♥ ①Ðt ❝➳❝ sè ❤➵♥❣ ❜❐❝ ♥❤✃t t❤❡♦ du✱ dv t❤× ❝ã t❤Ó ❧✃② ❣➬♥ ➤ó♥❣ ❝➳❝ ➤✐Ó♠✿ B 1 [x, y], B 2 [x + ∂x ∂u du, y + ∂x ∂u du] B 3 [x + ∂x ∂u du + ∂x ∂v dv, y + ∂y ∂u du + ∂y ∂v dv] B 4 [x + ∂x ∂v dv, y + ∂x ∂v dv] ❱í✐ ➤é ❝❤Ý♥❤ ①➳❝ ❜Ð✱ ❜❐❝ ❝❛♦✱ tø ❣✐➳❝ B 1 B 2 B 3 B 4 ❧➭ ♠ét ❤×♥❤ ❜×♥❤ ❤➭♥❤✳ ❉✐Ö♥ tÝ❝❤ ❝ñ❛ ♥ã ❣✃♣ ➤➠✐ ❞✐Ö♥ tÝ❝❤ ❝ñ❛ t❛♠ ❣✐➳❝ B 1 B 2 B 3 ✳ ❚õ ❤×♥❤ ❤ä❝ ❣✐➯ tÝ❝❤✱ t❛ ➤➲ ❜✐Õt r➺♥❣ ❤❛✐ ❧➬♥ ❞✐Ö♥ tÝ❝❤ t❛♠ ❣✐➳❝ ❝ã ❝➳❝ ➤Ø♥❤ t➵✐ ❝➳❝ ➤✐Ó♠ (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) ❜➺♥❣ ❣Ý trÞ t✉②Öt ➤è✐ ❝ñ❛     x 2 − x 1 x 3 − x 2 y 2 − y 1 y 3 − y 2     ➳♣ ❞ô♥❣ ❝➠♥❣ t❤ø❝ ➤ã ❝❤♦ tr➢ê♥❣ ❤î♣ ❝ñ❛ t❛✱ t❛ ❝ã✿ ✈í✐ ➤é ❝❤Ý♥❤ ①➳❝ ➤Õ♥ ❝➳❝ ➤é ❜Ð ❜❐❝ ❝❛♦ ❤➡♥ t❤× S =     ∂x ∂u du ∂x ∂v dv ∂x ∂u du ∂y ∂v dv     =     ∂x ∂u ∂x ∂v ∂x ∂u ∂y ∂v     dudv ❱× ✈❐② I =  D f(x, y)dS =  D f[x(u, v), y(u, v)] |J|dudv ✶✳✹✳✷✳ ➜æ✐ ❜✐Õ♥ tr♦♥❣ tä❛ ➤é ❝ù❝ ❛✳ ➜æ✐ ❜✐Õ♥✿ D      x = r cos ϕ y = r sin ϕ ✈í✐ r ≥ 0; 0 ≤ ϕ < 2π ↔ D   r 1 (ϕ) ≤ r ≤ r 2 (ϕ) ϕ 1 ≤ ϕ ≤ ϕ 2 ✺ I =  D f(x, y)dxdy =  D  f(r cos ϕ, r sin ϕ) rdrdϕ = ϕ 2  ϕ 1 dϕ r 2 (ϕ)  r 1 (ϕ) f(r cos ϕ, r sin ϕ) rdr P❤Ð♣ ➤æ✐ ❜✐Õ♥ ♥➭② t❤➢ê♥❣ ❞ï♥❣ ❦❤✐ ❜✐➟♥ ❝ñ❛ D ❝❤ø❛ x 2 + y 2 ✳ ❜✳ ➜æ✐ ❜✐Õ♥ ✭❚♦➵ ➤é ❝ù❝ tÞ♥❤ t✐Õ♥✮  x = a + r cos ϕ y = b + r sin ϕ ✱ ✈í✐ r ≥ 0; 0 ≤ ϕ < 2π ❑❤✐ ➤ã J = r ✈➭ (x − a) 2 + (x − b) 2 = r 2 ✳ P❤Ð♣ ➤æ✐ ❜✐Õ♥ ♥➭② t❤➢ê♥❣ ❞ï♥❣ ❦❤✐ ❣➷♣ ❜✐Ó✉ t❤ø❝ (x − a) 2 + (x − b) 2 ✳ ➜æ✐ ❜✐Õ♥ ✭t♦➵ ➤é ❝ù❝ ❝♦ ❣✐➲♥✮  x = ar cos ϕ y = br sin ϕ ✱ ✈í✐ r ≥ 0; 0 ≤ ϕ < 2π ❑❤✐ ➤ã J = abr ✈➭ x 2 a 2 + y 2 b 2 = r 2 ✳ P❤Ð♣ ➤æ✐ ❜✐Õ♥ ♥➭② t❤➢ê♥❣ ❞ï♥❣ ❦❤✐ ❜✐➟♥ ❝ñ❛ D ❧➭ ♠✐Ò♥ ❡❧Ý♣✳ ✶✳✻✳ ø♥❣ ❞ô♥❣ ❤×♥❤ ❤ä❝ ❝ñ❛ tÝ❝❤ ♣❤➞♥ ❦Ð♣ ✶✳✻✳✶✳ ❚Ý♥❤ ❞✐Ö♥ tÝ❝❤ ♠✐Ò♥ ♣❤➻♥❣ ❈➠♥❣ t❤ø❝ S =  D dxdy ✶✳✻✳✷✳ ❚Ý♥❤ t❤Ó tÝ❝❤ ❝ñ❛ ✈❐t t❤Ó ❛✳ ❍×♥❤ trô ❝♦♥❣ V      P❤Ý❛ tr➟♥✿ z = f(x, y) ∈ C(D) P❤Ý❛ ❞➢í✐✿ z = 0 ❳✉♥❣ q✉❛♥❤✿ ➤➢ê♥❣ s✐♥❤ ✴✴ Oz tù❛ tr➟♥ ❜✐➟♥ D✳ ❈➠♥❣ t❤ø❝ S =  D f(x, y)dxdy ❜✳ ❈❤ó ý • ◆Õ✉ ♠✐Ò♥ ❱ ❝ã tÝ♥❤ ➤è✐ ①ø♥❣ t❤× ❝ã t❤Ó tÝ♥❤ t❤Ó tÝ❝❤ ♠ét ♣❤➬♥ rå✐ s✉② r❛ t♦➭♥ t❤Ó ✻ • ◆Õ✉ ✈❐t t❤Ó V ❧➭ ❤×♥❤ t✉ú ý • ◆Õ✉ ❤×♥❤ trô ❝♦♥❣ ❝ã ❝➳❝ ➤➢ê♥❣ s✐♥❤ s♦♥❣ s♦♥❣ ✈í✐ trô❝ 0x ❤♦➷❝ 0y t❤× ➤æ✐ ✈❛✐ trß ❝ñ❛ x ✈í✐ z ❤♦➷❝ y ✈í✐ z tr♦♥❣ ❝➳❝ ❝➠♥❣ t❤ø❝ tr➟♥✳ ❇❚ ✻✳✼✳✷✳ ❉♦ tÝ♥❤ ➤è✐ ①ø♥❣✱ ❝❤Ø ❝➬♥ tÝ♥❤ 1 4 t❤Ó tÝ❝❤ ❝➬♥ t×♠ tr♦♥❣ ♣❤➬♥ t➳♠ t❤ø ♥❤✃t✳ ❑❤✐ ➤ã ♠➷t ♣❤Ý❛ tr➟♥ ❧➭ x 2 +y 2 +z 2 = 4a 2 ❤❛② z =  4a 2 − x 2 − y 2 ✳ ▼➷t ❞➢í✐ x 2 + y 2 = 2ay ❤❛② x 2 + (y − a) 2 = a 2 ✳ V = 1 4  D  4a 2 − x 2 − y 2 dxdy ✶✳✻✳✸✳ ❚Ý♥❤ ❞✐Ö♥ tÝ❝❤ ♠➷t ❝♦♥❣ ▼➷t ❝♦♥❣ z = f(x, y) ❝ã ❤×♥❤ ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝ tr➟♥ ♠➷t ♣❤➻♥❣ Oxy ❧➭ ♠✐Ò♥ D ✈➭ f(x, y), f  x , f  y ∈ C(D) t❤× S =  D  1 + (z  x ) 2 + (z  y ) 2 dxdy ❈➳❝❤ ❣✐➯✐ t❤Ý❝❤✿ ❚❛ ➤Þ♥❤ ♥❣❤Ü❛ ❞✐Ö♥ tÝ❝❤ S ❝ñ❛ ♠➷t ➤ã ♥❤➢ ❣✐í✐ ❤➵♥ S = lim S n = lim n  i=1 SS i ✈í✐ ➤➢ê♥❣ ❦Ý♥❤ ❝ñ❛ ❝➳❝ ♣❤➬♥ tö SS i ❝ò♥❣ ♥❤➢ ➤➢ê♥❣ ❦Ý♥❤ ❝ñ❛ ❝➳❝ ♣❤➬♥ tö D i ❞➬♥ tí✐ 0✳ SS i ❧➭ ❝➳❝ ❤×♥❤ ♣❤➻♥❣ ❝ã ❤×♥❤ ❝❤✐Õ✉ trù❝ ❣✐❛♦ ❧➭ D i t❤× D i = SS i | cos α i | α i ❣ã❝ ❝ñ❛ ♣❤➳♣ t✉②Õ♥ ❝ñ❛ ♠➷t SS i ✈í✐ trô❝ z✳ ◆➟♥ S = lim S n = lim n  i=1 SS i = lim n  i=1 1 | cos α i | D i = = lim n  i=1  1 + (z  x ) 2 + (z  y ) 2 D i =  D  1 + (z  x ) 2 + (z  y ) 2 dxdy ❇❚ ✻✳✽✳✶✳ ❛✮ P❤➬♥ ❞✐Ö♥ tÝ❝❤ ♠➷t ❝➬✉ ♥➺♠ ❜➟♥ tr♦♥❣ ♠➷t trô✿ ✼ tí ố ứ ỉ ét t tr t tứ t ó z = a 2 x 2 y 2 z x = x a 2 x 2 y 2 z y = y a 2 x 2 y 2 1 + (z x ) 2 + (z y ) 2 = a a 2 x 2 y 2 S = 4 D a a 2 x 2 y 2 dxdy ớ ề tí D ử ì trò x 2 + y 2 = ay tr ó t tứ t ủ t Oxy ù t ộ ự x = r cos , y = r sin S = 4a 2 0 d a sin 0 rdr a 2 r 2 Đ í ộ tí ộ số u = f(x, y, z) ị tr ề V ó ị ủ Oxyz V ột tỳ ý t n ề ỏ 1 , 2 , ., n r ỗ S i ột ể tỳ ý M i (x i , y i , z i ) ổ I n = n i=1 f(x i , y i , z i )V ( i ) ợ ọ tổ tí ủ số f(x, y) tr ề D ọ d i = ờ í ủ i sup d(M, M ), M, M i ế n s max{d i } 0 I n tớ ột ớ ữ I ụ tộ ề V ể M i tr ỗ i tì ớ ợ ọ tí ộ ủ số f(x, y) tr ề V ý ệ V f(x, y)dV := lim max{d i }0 I n V ✿ ♠✐Ò♥ ❧✃② tÝ❝❤ ♣❤➞♥ f✿ ❤➭♠ ❞➢í✐ ❞✃✉ tÝ❝❤ ♣❤➞♥  D f(x, y)dV tå♥ t➵✐✱ t❛ ♥ã✐ f(x, y, z) ❦❤➯ tÝ❝❤ tr♦♥❣ ♠✐Ò♥ V ✳ ❱× tÝ❝❤ ♣❤➞♥ ❦Ð♣ ❦❤➠♥❣ ♣❤ô t❤✉é❝ ❝➳❝❤ ❝❤✐❛ ♠✐Ò♥ V t❤➭♥❤ ❝➳❝ ♠➯♥❤ ♥❤á ♥➟♥ t❛ ❝ã t❤Ó ❝❤✐❛ V ❜ë✐ ❜❛ ❤ä ➤➢ê♥❣ t❤➻♥❣ s♦♥❣ s♦♥❣ ✈í✐ ❝➳❝ trô❝ t♦➵ ➤é✳ ❉♦ ➤ã dV = dxdydz ✈➭ ❝ã t❤Ó ✈✐Õt  V f(x, y)dV =  V f(x, y)dxdydz • ❈❤ó ý✿ ◆Õ✉ f(x, y, z) ❧✐➟♥ tô❝ tr♦♥❣ ♠✐Ò♥ ➤ã♥❣✱ ❜Þ ❝❤➷♥ V t❤× ❦❤➯ tÝ❝❤ tr♦♥❣ ♠✐Ò♥ ➤✃②✳ ✷✳✷✳ ❚Ý♥❤ ❝❤✃t✲❈➳❝❤ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❜é✐ ❜❛ tr♦♥❣ t♦➵ ➤é ✈✉➠♥❣ ❣ã❝ ❈➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ tÝ❝❤ ♣❤➞♥ ✸ ❧í♣ ❣✐è♥❣ ♥❤➢ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ tÝ❝❤ ♣❤➞♥ ✷ ❧í♣✳ ✷✳✸✳ ❈➳❝❤ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❜é✐ ❜❛ tr♦♥❣ ❤Ö t♦➵ ➤é ➜Ò ❝➳❝ ●✐➯ t❤✐Õt tå♥ t➵✐ tÝ❝❤ ♣❤➞♥ ✸ ❧í♣ I =  V f(x, y, z)dxdydz ❈➬♥ tÝ♥❤ I ✈í✐ ♠✐Ò♥ V ♥❤➢ s❛✉✿ ✶✮ ❱      a ≤ x ≤ b c ≤ y ≤ d g ≤ z ≤ h ✳ ❑❤✐ ➤ã✿ I = b  a    d  c   h  g f(x, y, z)dz   dy    dx = b  a dx d  c dy h  g f(x, y, z)dz ❍♦➷❝ t❛ ❝ã t❤Ó ➤æ✐ t❤ø tù ❧✃② tÝ❝❤ ♣❤➞♥✳ ✷✮ ❱      a ≤ x ≤ b y 1 (x) ≤ y ≤ y 2 (x) z 1 (x, y) ≤ z ≤ z 2 (x, y) ✳ ❑❤✐ ➤ã✿ I = b  a dx y 2 (x)  y 1 (x) dy z 2 (x,y)  z 1 (x,y) f(x, y, z)dz =  D dxdy z 2 (x,y)  z 1 (x,y) f(x, y, z)dz ✾ ❚r♦♥❣ ➤ã ❉  a ≤ x ≤ b y 1 (x) ≤ y ≤ y 2 (x) ✱ D ❧➭ ❤×♥❤ ❝❤✐Õ✉ ❝ñ❛ V ❧➟♥ ♠➷t ♣❤➻♥❣ Oxy✳ ❈❤ó ý✿ I =  V dxdydz ❧➭ ❝➠♥❣ t❤ø❝ ➤Ó tÝ♥❤ t❤Ó tÝ❝❤ ❦❤è✐ ✈❐t t❤Ó V ✳ ✷✳✹✳ ❚Ý❝❤ ♣❤➞♥ ♣❤ô t❤✉é❝ t❤❛♠ sè ✈í✐ ❝❐♥ ❧➭ ❤➺♥❣ sè ➜◆ ✶ ❈❤♦ ❢✭①✱t✮ ①➳❝ ➤Þ♥❤ tr➟♥ ❤×♥❤ ❝❤÷ ♥❤❐t D  a ≤ x ≤ b c ≤ t ≤ d ✳ ●✐➯ sö ♠ç✐ t ∈ [c, d], f(x, t) ❦❤➯ tÝ❝❤ t❤❡♦ ❜✐Õ♥ x ✈➭ I = b  a f(x, t)dx = I(t) ❧➭ ♠ét ❤➭♠ ❝ñ❛ t ①➳❝ ➤Þ♥❤ tr➟♥ [c, d]✳ ❱✃♥ ➤Ò ➤➷t r❛✿ ✈í✐ ➤✐Ò✉ ❦✐Ö♥ ♥➭♦ t❤× I(t) ❧✐➟♥ tô❝✱ ❦❤➯ ✈✐✱ ❤❛② ❦❤➯ tÝ❝❤ tr➟♥ ➤♦➵♥ [c, d]❄ ➜▲ ✶✿ ◆Õ✉ ❤➭♠ sè f(x, t) ❧✐➟♥ tô❝ tr➟♥ ❤×♥❤ ❝❤÷ ♥❤❐t D t❤× I(t) ❝ò♥❣ ❧✐➟♥ tô❝ tr➟♥ ➤ã ✈➭ lim t→t 0 ∈[c,d] I(t) = b  a lim t→t 0 f(x, t)dx ➜▲ ✷✿ ◆Õ✉ ♠ç✐ t ∈ [c, d], f(x, t) ∈ C[a, b] ✭t❤❡♦ ❜✐Õ♥ x✮✱ ✈➭ ♥Õ✉ f  t ∈ C(D) t❤× I(t) ❦❤➯ ✈✐ tr➟♥ [c, d] ✈➭ I  (t) = b  a f  t (x, t)dx ➜▲ ✸✿ ◆Õ✉ f(x, t) ∈ C(D) t❤× I(t) ❦❤➯ tÝ❝❤ tr➟♥ [c, d] ✈➭ d  c I(t) = d  c   b  a f(x, t)dx   dt = b  a   d  c f(x, t)dt   dx ✶✵