一元函数的定积分

����� ������ � ����� §4.1 ����������������� §4.1.1 ffflfiflffifl�fl fl! " 1 #%$%& y = f(x) ')(%* [a, b] +%,%-%.%/%021)35476 y = f(x), 8%6 x = a, x = b 9 y = 0 :<;>=fl?<@>Afl?flB%C S( @ 4.1). (DflE2FflGflHfl?<@7I%J%K%F L Hfl?<@>I ) # a = x0 < x1 < · · · < xn = b. M ∆xi = xi − xi−1 (i = 1, · · · , n) λn = max i=1,···,n ∆ti. N 8O6 x = xi(i = 0, · · · , n) PQ4SROTOAOUO= n VOWOXOY ( @ 4.1). ZO[ ξi ∈ [xi−1, xi], \ 4>RflTflAfl?flBflCfl]fl^%_ n∑ i=1 v(τi)∆ti, `flaflb λn → 0 cfld2Hfleflfflgflhfldjilk%m%nflg%hflo%p)4>R%T%Afl?%BflC S. " 2 #flqflVflrflsflt%u5v%w%x v(t) ofl8fl6fluflvfl0y1%r5s%'%c)*2z [a, b] {>| } ?fl~fl s. € ∆ti = ti − ti−1 (i = 1, · · · , n), fl^flHfl‚fldjilkflmflgflh lim λn→0 n∑ i=1 v(ξi)∆ti oflpflrflsfl'flc<*>z [a, b] {>| } ?fl~fl s. ƒ sfl„<…>E 1. † λn → 0 cfld5(fl*>?flUfl‡flˆfleflK%Zfl‰%?%0 2. (fl*>Hfl[fls ξi Ł KflZfl‰fl?fl0 ‹ T : a = x0 < x1 < · · · < xn = b € ∆xi = xi − xi−1 (i = 1, · · · , n), Œ ‹ ‖T‖ = max i=1,···,n ∆xi pflUfl‡ T ?flflxfl0 1 ffflŽ # f(x) ' [a, b] fflflfl0 `flafl‘ & A ’fl“flE>”flZfl• ε > 0, f δ > 0, – — Ufl‡ T ’fl“ ‖T‖ < δ, \ ”flZfl˜ ξi ∈ [xi−1, xi] (i = 1, 2, · · · , n) Jflf∣∣∣∣∣ n∑ i=1 f(ξi)∆xi −A ∣∣∣∣∣ < ε, ™fl‹ f(x) ' [a, b] Riemann šflCfld ‹ A p f(x) ' [a, b] Hfl? Riemann CflUfl07€fl= ∫ b a f(x)dx = A. 'flfl<{>d2›fle S(T ) = n∑ i=1 f(ξi)∆xi œ f(x) ' [a, b] Hfl?flqflVflCflUfl›fl Riemann ›fl0>€flI ∫ b a f(x)dx {>d f(x) œ>ž C $fl&fld f(x)dx œ>ž CflŸfl flefl0 ” ∫ ” œ CflUflIfld a › b Ufl¡ œ CflUfl¢flhfl›flCflU%H hfl0 " 1 # f(x) = c Kfl£fl¤fl$fl&fl0 \ 'flZ%˜<(%* [a, b] Hfld f(x) šflCfld2Œflf ∫ b a f(x)dx = c(b− a). §4.1.2 ¥flfifl¦fl§fl¨ fffl© 1 # f(x) ' [a, b] HflšflCfld \ f(x) ªfl' [a, b] ffl«fl0 ¬fl­fl® kfld ` ¢fl‚fl0 " 1 1fl¯flE Dirichlet $fl&fl0 D(x) = { 1, x ∈ Q; 0, x∈ Q ' [0, 1] H ® šflCfl0 " 2 # f(x) ' [a, b] š%C%d g(x) ' [a, b] f%%%d2Œ%-%° x = x0 t%±%d f(x) = g(x) .fl=fl²fl0 \ g(x) Ł ' [a, b] šflCfld2Œflf ∫ b a g = ∫ b a f. 2 fffl© 2 (1) # f(x) ' [a, b] Hflffl«fld2Œfl-fl°flfflhflVfls%t%±fld f(x) JflKfl+fl, ?fld \ f(x) ' [a, b] šflCfl0 (2) # f(x) ' [a, b] Hfl³fl´fld \ f(x) ' [a, b] šflCfl0 fffl© 3 (Newton-Leibniz) # f(x) ' [a, b] šflCfld2Œflfflµfl$fl& Φ(x), \ ∫ b a f = Φ(b)− Φ(a) = Φ(x) ∣∣∣∣ b a . ¶ # T : a = x0 < x1 < · · · < xn = b K [a, b] ?flZflqflVflUfl‡fld \ 3>·flU){7¤fl%¸%šfl¹%f ξi ∈ (xi−1, xi) º Φ(b)− Φ(a) = n∑ i=1 [Φ(xi)− Φ(xi−1)] = n∑ i=1 Φ′(ξi)∆xi = n∑ i=1 f(ξi)∆xi. » k¼H¼e¼½fl¾%KflqflV%¿flÀ%? Riemann ›¼0Á3 f(x) ?¼š¼C¼Â¼dÃM ‖T‖ → 0, ™¼Ä¼Å∫ b a f = Φ(b)− Φ(a). Newton-Leibniz Æfle ( fl¸ 3) Pfl1flflCflUfl?<Ç>ÈflÉflÊfl=fl1flµfl$%& ( ® flCflU ) ?<Ç>Èfl02'fl·flCflU<{>f “ ËflÌflfl¸ ” ­ ‹ 0 " 1 1 ∫ 2 1 dx x . " 2 1flgflh lim n→∞ [( 1 + 1 n )( 1 + 2 n ) · · · ( 1 + n n )] 1 n . Í M an = ln [( 1 + 1 n )( 1 + 2 n ) · · · ( 1 + n n )] 1 n = 1 n n∑ k=1 ln ( 1 + k n ) . μϼÐÒÑÔÓ d an K¼$¼& f(x) = ln(1 + x) ' [0, 1] H¼?¼q¼V¼CflUfl›fl0Õ3>_ ln(1 + x) šflCflŒflfflµfl$fl& (x+ 1) ln(1 + x)− x, Ö lim an = lim 1 n n∑ k=1 ln ( 1 + k n ) = ∫ 1 0 ln(1 + x)dx = [(1 + x) ln(1 + x)− x] ∣∣∣1 0 = 2 ln 2− 1. 3 :flt µfle = 4 e . §4.1.3 ffflfiflffifl�fl×flØ ff¼© 4 # f(x) › g(x) ' [a, b] š¼C¼d c1, c2 K¼£¼&¼d \ c1f(x) + c2g(x) Ł ' [a, b] šflCfld2Œflf ∫ b a (c1f + c2g)dx = c1 ∫ b a f + c2 ∫ b a g. nflfl¸fl„<…>flCflUflÙflf%6flÂ%Âflr%Ú fffl© 5 # f(x) › g(x) ' [a, b] šflCfld2' [a, b] H f(x) ≥ g(x), \∫ b a f ≥ ∫ b a g. nflfl¸fl„<…>flCflUflÙflf%ÛflÜ%Âfl0 ÝflÞ `fla f(x) ' [a, b] šflCfld2' [a, b] H m ≤ f(x) ≤M , \ m(b− a) ≤ ∫ b a f ≤M(b− a). n ®flß efl£ N Óflàflá %CflU%?fl¤%Ú fffl© 6 # f(x) ' [a, b] šflCfld \ |f(x)| Ł ' [a, b] šflCfl0 ÝflÞ `fla f(x) ' [a, b] šflCfld \ ∣∣∣∣ ∫ b a f ∣∣∣∣ ≤ ∫ b a |f |. fffl© 7 # f(x) › g(x) ' [a, b] HflšflCfld \ f(x)g(x) ' [a, b] šflCfl0 ff¼© 8 `¼a f(x) ' [a, b] š¼C¼d \ ”¼Z¼˜ a ≤ c < d ≤ b, f(x) ' [c, d] š¼C¼0 fffl© 9 # a < c < b, `fla f(x) ' [a, c] › [c, b] JflšflCfld \ f(x) ' [a, b] š Cfld2Œflf ∫ b a f = ∫ c a f + ∫ b c f. fffl© 10 `fla f(x) ' [α, β] šflCfld \ ” [α, β] {>?flZfl‰flâfls a, b, c Jflf∫ b a f = ∫ c a f + ∫ b c f. 4 fffl© 11 # f(x) ' [a, b] +fl,fl- ®flã Ifl02ä ∫ b a f = 0, \ f(x) ≡ 0. fffl© 12 ( åflæ�çlèfléflê ) # f(x) ' [a, b] +fl,fld \ ªflf ξ ∈ (a, b), º ∫ b a f = f(ξ)(b− a). fl¸ 12 f<… » ?flëfl˜fl‰fl ( ì<@ 4.2). í [a, b] Hfl?<4>RflTflAfl?flBflCflªflî [a, b] Hfl?flïflVflðflAflBflCflñ ß 0 5 §4.2 ò�����ó�ô���õ fffl© 1 # f(x) ' [a, b] šflCfld2' x0(x0 ∈ [a, b]) +fl,fld \ F (x) = ∫ x a f ' x = x0 šfl·fld2Œflf F ′(x0) = f(x0). ¶ [ ∆x º x0 + ∆x ∈ [a, b], \ f F (x0 + ∆x)− F (x0) ∆x − f(x0) = 1 ∆x ∫ x0+∆x x0 (f(x)− f(x0))dx. 3>_ f(x) ' x0 +fl,fld2Öfl”flZfl• ε > 0, öfl' δ > 0, º b |∆x| < δ cflf |f(x)− f(x0)| < ε, í b |∆x| < δ cfld2f ∣∣∣∣F (x0 + ∆x)− F (x0)∆x − f(x0) ∣∣∣∣ < ε. :flt F ′(x0) = lim ∆x→0 F (x0 + ∆x)− F (x0) ∆x = f(x0). fffl© 2 # f(x) ' [a, b] Hfl+fl,fld \ F (x) = ∫ x a f(t)dt, x ∈ [a, b] K f(x) ?flqflVflµfl$fl&fl0 nflfl¸fl„<…>Zfl˜fl'<(fl*7Hfl+%,%?fl$%&%Jflf%µfl$%&%öfl'�÷ ff¼© 3 (Newton-Leibniz øÒù ) # f(x) ' [a, b] +¼,¼d Φ(x) K f(x) ' [a, b] ? ZflqflVflµfl$fl&fld \ f∫ x a f(t)dt = Φ(x)− Φ(a) = Φ(t) ∣∣∣∣ x a (a ≤ x ≤ b). ¶ 3>fl¸ 4.1.3 í Ä 0 ¿fl¡flf ∫ b a f = Φ(b)− Φ(a) = Φ(x) ∣∣∣∣ b a . 6 37_%%¸ 2 ›%%¸ 3 ` n%ú%û%ü%ý%þ%ß%·%U%›%C5U��5q%”5”%²5B%?��%x�� qflÂfl0����� flP� �� œ�� “ ·flCflUflËflÌflfl¸ ”. " 1 1 I = ∫ 3 −2 max(x, x2 − 2)dx. " 2 # F (x) = ∫ ϕ(x) a f(t)dt. 1 F ′(x). Í M G(u) = ∫ u a f(t)dt, \ F (x) = G(ϕ(x)). 3� ��fl$fl&fl?fl1���� \ ™%ÄflÅ F ′(x) = G′(ϕ(x))ϕ′(x) = f(ϕ(x))ϕ′(x). " 3 # F (x) = ∫ ψ(x) ϕ(x) f(t)dt, 1 F ′(x). Í F (x) = ∫ a ϕ(x) f(t)dt+ ∫ ψ(x) a f(t)dt = ∫ ψ(x) a f(t)dt− ∫ ϕ(x) a f(t)dt. :flt F ′(x) = f(ψ(x))ψ′(x)− f(ϕ(x))ϕ′(x). 7 §4.3 ������������������������� §4.3.1 ffflfiflffifl������ff # F (x) K f(x) ' [c, d] Hfl?flqflVflµ%$%&50 3� ��5$%&5?%1���� \ š5¹%d b t ∈ [α, β]( [β, α]) cflf d dt F (ϕ(t)) = F ′(ϕ(t))ϕ′(t) = f(ϕ(t))ϕ′(t) ƒ RflCflU�fi<3 Newton-Leibniz Æfle í f fffl© 1 # f(x) ' [c, d] +fl,fld [a, b] ⊂ [c, d]. `fla x = ϕ(t) ' [α, β](  [β, α]) ffl+fl,��fl&fld ϕ(α) = a, ϕ(β) = b, - b t ∈ [α, β] (  [β, α]) cfld ϕ(t) ∈ [c, d]. \ ∫ b a f = ∫ β α f(ϕ(t))ϕ′(t)dt. fl�ffi CflU��fl?��flsflE 1◦ ®� —�!�"�# fl K�$flf ¬ $%&fl?)Ç>È�% 2◦ ®� —�&�' P ¬ $fl& t = ϕ−1(x) #)(�* d2– 8�+ á�, F (ϕ(t)) ∣∣∣β α ™�- ßfl0 " 1 1 ∫ a 0 √ a2 − x2dx (a > 0). Í M x = a sin t. t = 0 cfld x = 0, t = pi2 cfld x = a. Ö∫ a 0 √ a2 − x2dx = a2 ∫ pi 2 0 cos2 tdt = a2 2 ∫ pi 2 0 (1 + cos 2t)dt = pia2 4 + a2 4 sin 2t ∣∣∣∣ pi 2 0 = pia2 4 . " 2 1 ∫ 2 1 lnx x dx. Í ∫ 2 1 lnx x dx = ∫ 2 1 lnxd lnx = 1 2 ln2 x ∣∣∣2 1 = 1 2 ln2 2. 8 " 3 # a > 0, 1 ∫ a 0 dx (a2 + x2)3/2 . " 4 # m K�.�/�0fl&fld21fl¯flE ∫ pi 2 0 sinm xdx = ∫ pi 2 0 cosm xdx. " 5 # f(x) K¼' (−∞,+∞) H¼+¼,¼dSŒflt l p2123¼?¼$fl&fl0Ô1%¯flEԔ%Zfl˜ ‘ & a, Jflf ∫ a+l a f = ∫ l 0 f. ¶ ���flf ∫ a+l a f = (∫ 0 a + ∫ l 0 + ∫ l+a l ) f, 'fl½fl¾�4flâflVflCflU<{>M x = t+ l, ™flÄflÅ ∫ l+a l f = ∫ a 0 f(t+ l)dt = ∫ a 0 f, :flt ∫ a+l a f = (∫ 0 a + ∫ l 0 + ∫ a 0 ) f = ∫ l 0 f. ‚ 5 „<…>d2ä f(x) t l p�1�3fld \ 'flZ%˜%qflV�5%x%p l ?<(fl*>Hfld f(x) ? CflUflJflKflq�6fl?fl0 §4.3.2 ffflfiflffifl�flffi�7flfiflffi�ff Dfl‰ Å (uv)′ = u′v + uv′, fi ƒ R�8 a Å b CflUfld í f fffl© 2 # u(x) › v(x) ' [a, b] Hflfflq�9fl+fl,��fl&fld \ f∫ b a uv′ = uv ∣∣∣b a − ∫ b a u′v, �:fl= ∫ b a udv = uv ∣∣∣b a − ∫ b a vdu. " 1 1 ∫ b a (b− x)(x− a)dx. 9 " 2 # m K�.�/�0fl&fld21 Im = ∫ pi 2 0 sinm xdx = ∫ pi 2 0 cosm xdx. Í I0 = pi 2 , I1 = 1. ¢flBfl# m ≥ 2, 3>U�;flCflU��flšfl¹ Im = ∫ pi 2 0 sinm xdx = − ∫ pi 2 0 sinm−1 xd cos x = − sinm−1 x cos x ∣∣∣pi2 0 + ∫ pi 2 0 cosxd sinm−1 x = ∫ pi 2 0 (m− 1) cos2 x sinm−2 xdx = (m− 1)Im−2 − (m− 1)Im. _flK ÄflÅ Im ?�<�=flÆfle Im = m− 1 m Im−2 (m ≥ 2). 1◦ m = 2n (n ≥ 1) cfld ™ f I2n = 2n− 1 2n I2n−2 = · · · = (2n− 1)!! (2n)!! pi 2 , e<{>d (2n)!! Ÿflþ ®�> } 2n ?fl:flffl/�?fl&fl?�@flCfld (2n− 1)!! Ÿflþ ®�> } 2n− 1 ?fl:flffl/�Afl&fl?�@flCfl0 2◦ m = 2n+ 1 (n ≥ 1) cfld2fl^flš Ä I2n+1 = (2n)!! (2n+ 1)!! I1 = (2n)!! (2n+ 1)!! . 10 *§4.4 ����B�C�D�E §4.4.1 F�G�ff # f(x) ' [a, b] Hflffl+fl,fl?�H�9��fl&fl0 P<(fl* n ß Ufld ÄflÅ U�� T : a = x0 < x1 < · · · < xn = b, IflN Taylor Æfle ÎflÏ ¯<… ∫ b a f = b− a n n∑ i=1 yi + (b− a)2 24n2 f ′′(ξ) (ξ ∈ [α, β]). # M2 K |f ′′(x)| ' [α, β] HO? &KJ ¤OdL�K� ™ ¹KM¼ð¼A¼ÆOe¼?2N2O ®2> } (b− a)3 24n2 M2. §4.4.2 P�G�ff # f(x) ' [a, b] Hflf�H�9fl+fl,��fl&fl02P)(%* n ß Ufl0 IflN Taylor Æfle ÎflÏ ¯<… ∫ b a f = b− a n n∑ i=1 ( n−1∑ i=1 yi + y0 + yn 2 ) − (b− a) 3 12n2 f ′′(ξ) §4.4.3 Q�R�S�ff (Simpson T�U ) P [a, b] ß Ufl= n V<(fl*>d�VflH�W%V)(fl*7?){7s%d ™ P [a, b] Ufl= 2n V ß 5 ?<(fl*>02#���XflUflsflK a = x0 < x1 < x2 < · · · < x2n−1 < x2n = b. M yk = f(xk), ”�WflV<(fl* [x2k−2, x2k] (k = 1, 2, · · · , n), } Ak(x2k−2, y2k−2), Bk(x2k−1, y2k−1),Ck(x2k, y2k) o�Y�Zfl6fld `fla f(x) ' [a, b] Hflffl+fl,fl?)[�9��fl&fld \ šfltfl¯<… ∫ b a f = b− a 6n ( y0 + y2n + 4 n∑ k=1 y2k−1 + 2 n−1∑ k=1 y2k ) − (b− a) 5f (4)(ξ) 180(2n)4 , 11 §4.5 ������\�]�^�_ §4.5.1 `���ff a } ” · ffi � ” Ó¼á2, ï¼V¼U�bfl')(fl* [a, b] H¼?2c Q = Q([a, b]). — Q ? U�bfl’fl“fl¢flB ƒ VflY�d%0 4flq�efld�f Ñ�g ;fl· ffi c%?fl]%^%Ÿfl %e ∆Q ≈ q(x)∆x. – — �2�2h�i��flV%Ÿfl %eflK “ j2h¼? ” ]¼^¼d ™ š¼t%P� �k�:%= ( ¼82+2:¼= ) · ffi ß e dQ = q(x)dx. 4�H�efld2CflUflHflefl1 Ä�l cfl0 §4.5.2 m�n)o�Sfl��p�q #�r�s<4>6 L ?flˆflflK y = f(x) (a ≤ x ≤ b). 1�t�u�5fl0�4flq�efld�h%%· ffi u�5fl?%]%^flŸ% fle%0 ∆l ≈ √ ∆x2 + [f(x+ ∆x)− f(x)]2, 3>·flU<{>¤flÆflefl› f ′(x) ?fl+fl,flÂfld�u�5fl· ffi ( í u�5fl·flU ) p dl = √ 1 + f ′2(x)dx. Öflf l = ∫ b a √ 1 + f ′2(x)dx. " 1 L : y = ln cos x, 0 ≤ x ≤ pi6 . 1 L ?�u�5 l. Í l = ∫ pi 6 0 √ 1 + sin2 x cos2 x dx = ∫ pi 6 0 dx cos x = 1 2 ln 3. 12 #<4>6 L 3�vfl&flˆflflE { x = ϕ(t), y = ψ(t) (α ≤ t ≤ β) • Ñ d b ϕ′(t) > 0 c¼dlM a = ϕ(α), b = ϕ(β), x = ϕ(t) ö¼' ¬ $¼& t = ϕ−1(x) (x ∈ [a, b]), vfl&flˆfl ™ hfl [a, b] Hfl?fl$fl& y = ψ(ϕ−1(x)), _flK ™ f l = ∫ b a √ 1 + ( dy dx )2 dx = ∫ b a √ 1 + ψ′2(t) ϕ′2(t) dx, o ã c # fl x = ϕ(t), ™flÄflÅ l = ∫ β α √ ϕ′2(t) + ψ′2(t)dt. (1) μÏ2w ¯¼d b ϕ′(t) < 0 c¼d M a = ϕ(β), b = ϕ(α), \ H¼B¼?¼e2x Ł Kfl=fl²fl?fl0 " 2 1�y�zfl6flq�{ { x = a(t− sin t), y = a(1− cos t) 0 ≤ t ≤ 2pi ?�u�5fl0 Í l = ∫ 2pi 0 a2 √ (1− cos t)2 + sin2 tdt = √ 2a ∫ 2pi 0 √ 1− cos tdt = 2a ∫ 2pi 0 sin t 2 dt = 8a. `fla 4>6 L 3>g�|�}flˆfl r = r(θ) (α ≤ θ ≤ β) • Ñ 0 ™ ñ b _ L 3�vfl&flˆfl{ x = r(θ) cos θ, y = r(θ) sin θ, α ≤ θ ≤ β • Ñ 0��flc ÎflÏ�,<Ñ l = ∫ β α √ r2(θ) + r′2(θ)dθ. 13 " 3 1�~flAfl6 r = a(1 + cos θ) (0 ≤ θ ≤ 2pi) ?��5fl0 §4.5.3 m�n�€�Gfl��nflfi 1) 4>RflTflAfl?flBflC flÐ 3 x = a, x = b(a < b), y = 0 › y = f(x) ;>=fl?<4>RflTflAfl?flBflC S. 3fl@ 4.10 šfl¹ S = ∫ b a f K<4>RflTflAfl? # & ( šfl/flš�/fl? ) BflCfld�‚� fl?flëfl˜flBflC \ K S = ∫ b a |f |. `fla 4>RflTflAfl?�ƒflRflK�vfl&)476 L : { x = ϕ(t), y = ψ(t) (α ≤ t ≤ β, ϕ(α) = a, ϕ(β) = b). „  ϕ′(t) › ψ′(t) J%' [α, β] +%,%d2Œ%-%' (α, β) { ® p 0, \ x = ϕ(t) ' [α, β] f ¬ $fl& t = ϕ−1(x), 4>6flˆfl ™ šfltflŸ%=)(%* [a, b]  [b, a] Hfl? » $fl&flA e y = ψ(ϕ−1(x)). _flK<3 x = a, x = b, y = 0 › L ;>=fl?<4>RflTflAfl? # &%BflC ™ K S = ∫ b a ψ(ϕ−1(x))dx. fi N ã c # fl x = ϕ(t), ™flÄflÅ S = ∫ β α ϕ′(t)ψ(t)dt, ‚<4>RflTflAfl?flëfl˜flBflC S = ∫ β α |ϕ′(t)ψ(t)|dt. 14 ¼^¼d `¼a ψ(α) = c, ψ(β) = d, ψ′(t) 6= 0, \ 3 y = c, y = d, x = 0 › L ;Ô=¼? @>Afl? # &flBflCfl›flëfl˜%BflC%Ufl¡%K S = ∫ β α ψ′(t)ϕ(t)dt › S = ∫ β α |ψ′(t)ϕ(t)|dt. " 1 1Ò3 x …¼›2y2z¼6flq�{flE x = a(t− sin t), y = a(1− cos t), 0 ≤ t ≤ 2pi : ;>=fl?<@>AflBflC S. 3 x = a, x = b(a < b), y = y1(x) › y = y2(x) (y2(x) ≥ y1(x)) ;>=fl?<@>AflBflC K¼t y = y2(x) p2ƒ¼R¼?Ò4ÔRflT%Afl? # &flBflCflî%t y = y1(x) p2ƒ¼R¼?Ò4ÔRflTflAfl? # &flBflC ­ O ( @ 4.11), í f S = ∫ b a (y2(x)− y1(x))dx. " 2 1<3 y = x2 › y = √ x ;>=fl?flBflC S. Í S = ∫ 1 0 ( √ x− x2)dx = 1 3 . 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 2) tflµflsflp�ƒflsfl?<4>R�†flA%?flB%C 15 #<4>6 L 3>g�|�}flˆfl r = r(θ), α ≤ θ ≤ β (0 < α− β ≤ 2pi), • Ñ 0 b θ 3 α ã Ê Å β cfld54>6flHfl?fl”�‡%s)3 A ã Å B( @ 4.12). ¢flB�� Ñ 3 OA,OB › L ;>=fl?flBflCfl0 3 θ = θ0, θ = θ0 + dθ › L ;>=fl?fl· ffi BflCfl]fl^%q%V�ˆ�‰flp r(θ0), Ł�~�‹flp dθ ?�†flAfld�tflBflCflp 12r 2(θ0)dθ. Öfl:fl1<4>R�†flAfl?flBflC%p S = 1 2 ∫ β α r2(θ)dθ. " 3 1�Œ�fl6 (x2 + y2)2 = a2(x2 − y2) (a > 0) ;>=fl?flBflC ( @ 4.13). -1 -0.5 0.5 1 -0.4 -0.2 0.2 0.4 Í 4>6fl?flg�|�}flˆflflp r = a √ cos 2θ. 3fl@>Afl?fl” ‹  ( @ 4.13) šfl¹fl:fl1flBflCflp S = 4 · 1 2 ∫ pi 4 0 a2 cos 2θdθ = a2. 3 θ = α, θ = β (0 < β −α ≤ 2pi), r = r1(θ) › r = r2(θ) (r1(θ) ≥ r2(θ)) ;Ô=¼? BflC ( @ 4.14) \ K S = 1 2 ∫ β α (r21(θ)− r22(θ))dθ. `fla r2(θ) = 0, ™ Êfl=<@ 4.12 ?�ŽflAfl0 §4.5.4 ��‘fl��‘flfi #flfflqflV�Z�’ V , } Ox …flHflqfls x oflî x …�“fl8fl?�”flBfld�• Ä Z�’fl?�•%B Cflp S(x). IflN · ffi � ™ šfltfl1 Ä�– ' x › x+ dx ­ *>?fl· ffi ’flCflp ( @ 4.17) dV = S(x)dx. 16 Ö�Z�’ V – ' x = a › x = b ­ *>?�’flCflp V = ∫ b a S. ¿fl¡fld b Z�’flK<3fl4>6 L : y = f(x), y = 0, x = a › x = b ;>=fl?<4>RflTflA�— Ox …�yflÉfl: Ä c ( @ 4.17). 3>_ S(x) = pif 2(x), ™flÄflÅ�˜ yflÉ�’fl?�’flCflp V = pi ∫ b a f2 (1) `5a 426 L Kj3�v5&5ˆ5 x = ϕ(t), y = ψ(t) (α ≤ t ≤ β) • Ñ ?5d „  ϕ′(t) > 0, \ ñ b _ (1) o ã c # fl x = ϕ(t), š Ä V = pi ∫ β α ϕ′(t)ψ2(t)dt. (2) g�|�}flˆflflšflt b =�v%&<476 Ó�™�š 0 " 1 1 x 2 a2 + y2 b2 = 1 ;>=fl?�”flB<@>A�— Ox …�yflÉfl: Ä yflÉ�’fl?�’flC%0 " 2 12›¼A¼6 x 2 3 + y 2 3 = a 2 3 ;Ô=¼?2”¼B<@7A�— Ox …2y¼É¼: Ä y%É�’%?�’ Cfl0 §4.5.5 ��‘fl��œ�nflfi # S K<3 y = f(x), (a ≤ x ≤ b) — Ox …�yflÉfl: Ä yflÉflB ( @ 4.19). �ž� – ' x › x+dx ­ *Ô?¼· ffi B¼C ™ KflqflV)Ł�Ÿfl?� flBflCfl0¡Ł�Ÿfl?flH�¢�ˆ�‰flp |f(x)|, ¢�¢�ˆ�‰flp |f(x+ dx)|, ‚�£��flp dl, ��¤ dl K<4>6 L ' x î x+ dx ­ *>?�u�5 ffi�¥ 02:flt dS = pi(|f(x)|+ |f(x+ dx)|)dl ≈ 2pi|f(x)| √ 1 + f ′2(x)dx, _flK� flBflC S = 2pi ∫ b a |f(x)| √ 1 + f ′2(x)dx. `fla L K<3�vfl&flˆfl x = ϕ(t), y = ψ(t), α ≤ t ≤ β • Ñ ?fld��ž S = 2pi ∫ β α |ψ(t)| √ ϕ′2(t) + ψ′2(t)dt. 17 " 1�ˆ�‰flp R ?�¦fl?flŸflBflC S. §4.5.6 §�¨�©�ª�« " " 1 #¼q2¬2­2®¼' x …¼H¼d ƒ ¾¼s¼p a › b. ¯Ô¹2t2°¼x¼p ρ(x)(a ≤ x ≤ b). 1�¬�­fl?�±�~fl0 x¯ = ∫ b a xρ(x)dx∫ b a ρ(x)dx . " 2 ˆ�‰flp R ²fl?�ˆ�¦flA�³�´)µ>Dfl’flß�³fldÔ1%m�³�¶�·fl:%ofl?�¸ ( ³fl?)¹ ±flp 1 º / ² 3). Í [%%8�‹�|�}�» Oxy, ˆ¼Ł x2 + (y − R)2 = R2 (0 ≤ y ≤ R) — Oy …�y É%: Ä y%É%B ™ K�ˆ�‰%p R ²%?�ˆ�¦%B ( @ 4.20). º�³�½)3 y(0 ≤ y ≤ R) ¾�¿ y− dy :¼o¼?2¸¼d ™ KflP�³�½fl' y− dy › y ­ *Ô?2³2À2Á�¿��flx R :¼o¼?2¸¼d Ö dW ≈ pix2(R− y)dy = pi(2Ry − y2)(R− y)dy. CflUflHflefld ™flÄflÅ W = pi ∫ R 0 (2Ry − y2)(R− y)dy = pi 4 R4 ( º · ² ). 18 §4.6  à � � §4.6.1 Ä�Å�Æ�Ç�Èfl��ÉflŽflfiflffi # f(x) ' [a,+∞) Hflfflflfld2”flZfl˜ b ≥ a, f(x) ' [a, b] šflCfld `fla lim b→+∞ ∫ b a f öfl'fld \ ‹�Ê flCflU ∫ +∞ a f Ë�Ì ( flöfl' ), Œfl-fl€ ∫ +∞ a f = lim b→+∞ ∫ b a f. ¬fl­ d ‹�Ê flCflU ∫ +∞ a f Í�Î 0 fl^�Ïflšfltflfl Ê flC%U ∫ a −∞ f. `fla f(x) ' (−∞,+∞) fflflfld2”flZfl˜ α < β, f(x) ' [α, β] šflCfld \ fl∫ +∞ −∞ f = ∫ a −∞ f + ∫ +∞ a f. –¼f¼'¼H¼e¼½flR ƒ V Ê flCflUflJ Ë�Ì ?�Ž�Ðfl¢fldÒÑflRfl? Ê flC%U� flK Ë�Ì ?fl0 Î Ï ¯<…>d2CflU Ë�Ì î�$%9%CflU%?fl¤%î a ?�Ófl[�Ôflf�Õ�»fl0 " 1 ™�š ∫ +∞ a dx xp ? Ë�Ì Âfl0�t<{ a > 0, p pfl£fl&fl0 Í 1◦ b p 6= 1 cflf ∫ b a dx xp = 1 1− p(b 1−p − a1−p). :flt b p > 1 cfld2CflU Ë�Ì d2Œflf∫ +∞ a dx xp = lim b→+∞ 1 1− p(b 1−p − a1−p) = a 1−p p− 1 . 19 b p < 1 cfld2CflU Í�Î ( ¿ +∞). 2◦ b p = 1 c ∫ b a dx x = ln b− ln a. CflU ∫ +∞ a dx x Í�Î ( ¿ +∞). í µflCflU b p > 1 c Ë�Ì d b p ≤ 1 c Í�Î 0��flKflq%V�Ö ' £ N ?�× a dÙØ Ú�Û�Ü J�Ý Dfl‰fl0 " 2 1 ∫ +∞ −∞ x 1 + x2 dx. Í Þ p ∫ +∞ −∞ xdx 1 + x2 = ln √ 1 + x2 ∣∣∣∣ +∞ 0 = +∞. ÖflCflU ∫ +∞ −∞ x 1 + x2 dx Í�Î 0 D¼E n¼‚¼„Ò…Ô£fl%CflU<{>+fl,fl?�Afl$fl&fl'fl” ‹ (fl*>Hfl?flCflUflp�ßfl?�× š d ” _ Ê flCflUflŒ ® qflfl=%²�÷ " 3 1 ∫ +∞ 1 arctan x x3 dx. " 4 1 ∫ +∞ 0 dx (1 + x2)(1 + xα) . Í M x = tanu, \ ∫ +∞ 0 dx (1 + x2)(1 + xα) = ∫ pi 2 0 du 1 + tanα u u= pi 2 −t = ∫ pi 2 0 dt 1 + cotα t = ∫ pi 2 0 tanα tdt 1 + tanα t . _flKflf ∫ +∞ 0 dx (1 + x2)(1 + xα) = 1 2 (∫ pi 2 0 du 1 + tanα u + ∫ pi 2 0 tanα u 1 + tanα u du ) = pi 4 . " 5 1 ∫ pi 2 0 dx 1 + cos2 x . §4.6.2 àflfiflffi # f(x) ' (a, b] +fl,fld�á f(x) ' a ?fl½� �â�ã)µ�ä�å�æ�����ç�è�é∫ b a f = lim ε→0+ ∫ b a+ε f. 20 ê�ë�ì�í�î�ï�ð�ñ�ò ç�ó�ô�õ�ö ∫ b a f Ë�Ì ò�÷�ø ó�ô�õ�ö ∫ b a f Í�Î ò a ó�ù�õ�ö í ô�ú�æ û�ü b ý�ô�ú�þ ò�ß���� è�é�ô�õ�ö∫ b a f = lim ε→0+ ∫ b−ε a f. � c ∈ [a, b] ý f(x) í ô�ú�þ ò è�é∫ b a f = ∫ c a f + ∫ b c f, � � ë��������� ô�õ�ö� ��� �þ ò���������í ô�õ�ö��� �æ � 1 � ∫ b a dx (x− a)p (b > a, p > 0). � � p = 1 þ ò ∫ b a dx x− a = limε→0+ ∫ b a+ε dx x− a = lim ε→0+ ( ln(b− a) + ln 1 ε ) = +∞ � p 6= 1 þ ò ∫ b a dx (x− a)p = limε→0+ ∫ b a+ε dx (x− a)p = 1 1− p limε→0+(x− a) 1−p ∣∣∣∣ b a+ε = 1 1− p limε→0+((b− a) 1−p − ε1−p) = { 1 1− p(b− a) 1−p p < 1, +∞ p > 1. � p ≥ 1 þ ò ∫ b a dx (x− a)p 21 ��� æ � p < 1 þ ∫ b a dx (x− a)p = 1 1− p(b− a) 1−p. ��� õ�ö � p < 1 þ��� ò � p ≥ 1 þ ��� æ�� ß ý���ff�fi�fl í�ffi� ��� æ � 2 � ∫ 1 0 lnxdx. � 3 � ∫ 1 0 dx√ 1− x2 . §4.6.3 !�"�#�$�% Cauchy &�' (�)�*�+ A > 0, f(x) ñ [−A,A] � õ ( ,�-�.�é � õ ). /�0 lim A→+∞ ∫ A −A f ð�ñ�ò ç�ó�1�ù�.�é�õ�ö ∫ +∞ −∞ f í (Cauchy) 2�3�æ�4�5 lim A→+∞ ∫ A −A f = V.P. ∫ +∞ −∞ f. 6�7 ò � ∫ +∞ −∞ f �� �þ ò ç�8 ∫ +∞ −∞ f = V.P. ∫ +∞ −∞ f. ( a < c < b, � 0 < ε < c − a, 0 < ε < b − c þ ò f(x) ñ [a, c − ε] 9 [c + ε, b] � õ ( ,�.�é � õ ), /�0 lim ε→0+ (∫ c−ε a f + ∫ b c+ε f ) ð�ñ�ò ç�ó�1�ù�.�é�õ�ö ∫ b a f(x)dx í Cauchy 2�3 ò 4�5 lim ε→0+ (∫ c−ε a f + ∫ b c+ε f ) = V.P. ∫ b a f. : / ∫ 1 −1 dx x 22 4�;��� ò�< ý V.P. ∫ 1 −1 dx x = lim ε→0+ (∫ −ε −1 dx x + ∫ 1 ε dx x ) = lim ε→0+ (ln ε− ln ε) = 0. = / ∫ +∞ −∞ xdx ý ��� í�ò�< ý V.P. ∫ +∞ −∞ xdx = lim A→+∞ ∫ A −A xdx = 0. 23