Scribd
Upload
131 views29 pages

Integral Calculus PDF

Uploaded by

Cristele Mae Garcia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
131 views29 pages

Integral Calculus PDF

Uploaded by

Cristele Mae Garcia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
You are on page 1/ 29
INTEGRAL CALCULUS Caloulus “The piniges of integration uere Frmulated by Tome Newton and Gottfred Lelia. Ine late sevarteentn conury, rough the Ardawitha| tienen of cals, vid Hy werent ly doyered he dheorem demersal, the celafontp Veteen the wo Uriel operations of erlulus, diffenhati a wegetion. Igri tanned ui dvdownhation and He define kegel af a fandion ain We easly bonpated ove aw antidenakne is. Kou. “Re mdean wataton fir the definite gral ums teu by Gobi bre nS. Weald he ingasybo|,'S°) Fm an dhnya leer ¢ ; Saving fe Summa (Latin fr "sum" vr “total GENERAL FORM, one : FOR) isany Aarctin, sda Hat LMax = FEYEC | Floy > Foy) ond C is an Jo inkegtal sign array constant, Fix) > integrand C= tonstant of reg Fon) + C2 indefinite integral FORMULA The Power Formula” f u™ Jutdu = “Gr +0 = nel subscribe 0n youtube, Rf @enginerdmnatn ‘Logavithin Formula’ subserilbe 09 yystulve, Jo @enginerdmath Jae = Inu] +0 Je'du = e'+C Tha W Wa ulna . N — a Soatdu Je du ra ma Judy = uv- Jvdu note: wand V ave finchons of x a— js a anstant © = 8.71929 (bate of natwal logarithm) Te opty the Fiemula of integration lay paris, separate the unkgrand vio two aches yu and dv usually as the most vamp cated ‘ocor onntaining ax. ) J sinudu = ~ cosy +0 % JseCudus tanu + ) J agudu= sou tC 8) J secutanudy > secu te 3 Mtawdy = In secu $0 = ~Inasute 4) S$ otudu = Insinu +0 4) JS esctudu=-otu + C 5) Ssecudu~ \n (ecu ttanu FC f esoucctu = ~cscu te ¥) Sescudu = In[escu - cotu) +c subsorilbe,on yputuloe, Bb ‘noes GNING INVERSE TRIGONOMETRIC PUNCTING @enginerdmath _ it -| U 0S par a Tg FO jj o_ M1 gt Ub gy = o 3) [tu = awit + © a-u INTEGRATION BY TRIGONOMETRIC SuASTITUTION Three Cases May be enaluated: Tf the integrand mths : oz-u= , let U= asind ‘Cased. T the wlegrard ymohes Nour 5 lek w= otand TE the vwlegrard wnthes » uz=a* 4 tet u = asecd INTECRATON DF HYPEREOMO FUNCTIONS p Jsinhudy = ashutc 5) Ssechudu = sw" Canhy) +e 2) J msn du = sinh + Cy Seschudy = Wntanh’s +c 3) FS tarny dy = Inashy +e J cbhudu = \nanhu to Subscribe, 01 yputuloe, Bf DEFINITE INTEGRAL / WALLIS FORMULA Genginordinath z T L-i)(n-a)-.. or | [O-NUn3) x] Se ain" uas"udy = —————-, "x Cmanimtn-2) & vitor: y= Taf mand ate both even K= 1, otherwise WIGHER ORDER INTEGRALS ‘Veetated Double ategral b Ayo be A YUM) = ( 4, J, Veo dy drs ~ J, (Ss #en3)34) aK i) wate) xdy = f [Jr Fou ax] 4 Whee. =O Kio ate constants Foxy) iso Atoction of Rand y- “Hemled Tipe Integr) f eee cay addy dx YHY A KY) I. ere t Jona de Ay Yt a Jie gg) J % LY) S supsetibe on yputate 2.05 PLANE AREA IN RECTANGULAR COORDINATES @enginerdmath * he aica O= lower lint General Formato: =f ydx b= gpa lnc a = Weight xFa x= n= ditorential width Az So yax 3) Partly above and below the axis =a Ree AREA BERWEEN TWO CURVES b c A= “J yan J. ya wg) |A=S2G.-y) ax “J wae “y vor OK 6 D4, =f) eo Ye ace!2" Yaiontal ship Te wie [a= emday =~ th) J \ frag” %ea)dy x Subsarilve on yputulie, Rf Gengunerdiath T= vadws A= lower het P= veper ln AO= didhrential ange, OF the nuemental oreo, Length of 0 Plane Carve, 4 ds ed Zh a ‘Inteckangalae fxm: By Pyhagorean Theorem ds = 1+) go ) liebe ax or s= [ay dy Ine: 4 subserilee-07 youtube, Rb In Gengnerdmath bo S= f NV vaty + Colne gt yo, z so JP fies catasy ab Centroid of Plane brea: Case’: \orheal Stip haaSp nah Ay=JP yeah wie. Ah = y ax Je Since the Wnt is from Xi) te, engress ¥ in dorms of: Y= Yugger cine ~ Feuer one, \Cose2% Hoviaontal Ship a3 ag = fy Kea hg = SP ya px whee R= xd Gee He lint is fron, Ah, engss 61» Hovtns ey KE Arne ume — Xref ome, Pavatioic, Segment and” Spardte cobain wetutechi Genginerdmath For Parabola: Xe =2b Yer zh ita = 3-bh | For spandrel Z.-1 3, = 3 %e= yb] | e==bh A soni - $ bh Moment of Tnectiq of Plane Arcos Moment of mein aint the paris y Use ~vertical strip: y= J)" oda mars syd . * Moment of Inertiq obiowl the % ans Use haiental strip = S yaa Mee UA xy 4 Subsoribe, 09 yputuie, > Genginerdinath he: i. In saving Br He moment of inertia using wnegraton, dhe a stip & that all ements. OF the stvip will have Ihe same distame frum the ovis of manent Ge. honaontal ship Air moment abolt »-oAis and vertical strip BC moment’ olaut He y> anis) Forallel =oxis Theorems : (Transfer Inertia) [=a] hee: T, = cenboidal moment of inertia I= Lt Ad, paralld. f dhe wae Iy= Ty + Ady Ty = Cenboidal rome of ner ss povalel tb te YF ans. Noluwe of q Sdlid of Revolution ise Method yet Volume by Groalar dv= Tyodx Ve Tf yde whee y - Dagger Yor MINEMMONIC: One way of vemembercy His Aamula is Tosthink” of the sold Batry seed oni i nfintes ~ wally Hh deh of vadus J and Hanes ax, hon apf He Fala Avrplume which ts Th: Ty"dx lo Subseries, on youtube, Rb Cae: Nolume by tblow Cylndvica) Shel Gap ’ V= ae oy whee: XK a= age ~ Heese MNEMON\G: Dre wou of veneer Ti Reema ag * Anke alla cylindrical ae wath fe radasye jhaight h ard the nfeitesinally wall Anaboness dy. the femul Ae Naume of a byprerl Shell say be Shout obas aT rhar = aryxdy When an are of 0 Pare cunts tevehies about a lime yn te pave ot cating the arte the swface areq gewetatd is 4he yoducr Othe teryth of {his ave ard the aroumforene fF He dre traersed \y its cenvvoid, 3 - = are, A= arrg | whee & tf are 1 = diane Fon he centnid of the atc the ams of revolution ) The ~wuime of a solid of rewlution generated by a fone aiea which revels dt 0. ine ook sing H, is Re padck of eaten under tation ord he ciraunfaeree of te orcle, Jovared ‘by the centnid oF He slate avn, N subscribe 00 yprbube, RP Anis oF ronan Genginerdmatin / oF Ne wN = otra SS? where: Y= Yelume (= shocest distance fom the confraid tthe aiea ty We revolved ty the axis of vevolutivn Work by Inkegration =F. whee: W = uork wis F= umstant fie a= disdace ments ~The ‘nial work ame by a \orlalle Fite is axdx _ a aL re = \ ysu - dw an CK") MN = aA 5 2 * v = TSuudu ie 4K OK Cl-aeyt 4 is Bier lek ue Ite S Grants? lavtén) f= ay +e — * “ tin SUR) ft dose oe bx » 4h Purdy gu 6 2 ot 6-98) Of Nex dx Yo fy Gla: bet U= \+ Sun non daw ox) we Od xeury = S@-20 4)udu = Judy -2fur%edy 4 fa i 2 we ye ye = at + te Zam - £0, 2 (ne Fe 5 Subsorilbe on ypatulbe- Jf iD J SW 4 @engunerdinath a y, nk de Pz 2 +c tas lek w= HON? J 1% ae = au ages = S$ sinu @du) a =e OA = Baue e 2Sanudu 2 Sax Pecan A Stee (anx dx) Solin: ek U= 9x is A= SINAdX J ¥ = ghee LO = = (wax) +e We r S Sn cos 7id% J ud Soll: tet UF sinx du= wordy = a . = sintx * q lo = ain't _ ant lo) =i 4 4 = yu) - 410) “la “5 +-du 4 \n\ul +c 2: eau en +O subsofilte.0n yputulte, RPE x242 We) ; xtl ox @enginerdinath YolN: xe = Mey x | ne a pee Sy Sere Sect an) . = 5X xt a\njmal | = 2-24 3\no-3\n 3\n3- 30 alna = Ina” =| lng won (nx) a) SE Seth pp te atonal ie Solin. feb U= (nx Ane m = fudy ® ¥) co wn — = J esctndx _ Sond du ais Elnjoscu- at ul tC = & © ode 2 In| csc x~ atx + du LSescudu a . My 19) J Cesc4x ~ wk 4n)dx aly “a 4 Ji cecu~cotu au %, tn eacy -atul ~ Inlanul 4 & In | 6c 4x = coke} ~ {n\n \ . 1,4 J Som: fet us x du> 4dx au = ax § : u- otu) oY fsc 7 ‘le =|2 Ina. L 4 4, 4 stsribe sn ypube 6 Genginordinath ‘ $n fee oe ne aetnc t ue x =x" Sol: Set fet i) 19° te dye hax my as e"du aA we a) J nto nx a Sduution ! A 10% = fin” ax - ot - a wtus 3x SB) he d= 24x > 2S iotay 13 i co Sou =ax 3 ‘nlegrs ying In de ) TS - Soy: feb u= 3 J ( du = ase BAL dux 2d% ar = =a & * 3 ew 5 2 AS . : Seape = fx ~3xhbh)4is- So\'q: 55 xXte Sseea5r5 (x Sx hist % tet us OS Sothae (avy ttt Aus 0% ‘ = Shape 1. fog LES) ie yz” 3 td ka a sm Mom tC 4 TECHNIQUES DF INTEGRATION ssbearto on yulte, Rf @enginerdmath Inkegrabon By Farts 2) S almadx JSudv= we ®dy Sens let usm — thy=adx = (9) US) ~ Jon ave as = lox - 5S xdx * Bn ae te 7 PE lox - bone ®) J o*snxdx Sudv= we fidy Vn — Sota: bet we sink “ am Seamdx = compe) - J et warden P tosxX dx N= Use gration by ps agp esx let U= cox — dy> 70x dui ~ ~Sinxdx ye” Serax = eon ~ Leto — J 6% cam dx) Seramady = eX sme = 6% OK — lo Poinx gry Pfersindx = _C*sIK = © 5K Sehsinxdx = a ensnx - ye ax +c Imeagation of fouets of Sie ard Cosine, Case Ssntudy — o¢ f cos"4dy yuhere. M1 1S an edd_inkeger ah) p weadx = sine Syn: Pudrdrx = J ast Ccostdn)) ~ SI six) coord ; ds osc w 3 = J coxda- S sixunxdx - am — = snx —~ZE4c = AM" s 4C Is ssubserite on youtube, Re aa) f sm®xdx Gengmnerdinath Sein. ni and x =! fan’) sinnd% = S (\- wx) snx ox < PCy = 208% 4 cost rw dx = Panna - 26 o&xsimcdn + f cobs simad leh u= w5x — dus- sin«d x — aus sinKdx = farmed ~2Sur edu) + Fut Gav) = famdx + afte ~ falda = ~ 0K 4 2 we = “ te — 005K +B wer “LaFate Cose 2° F sins cos"x dx , Where atleast sie f the ppane a8) Sawx ask ax On ss xwsixdx = Sewrx cwstx sinndx -S Cr 05%) tosh sind cactxamrdx ~ f cos'x sinxdx —= cos’x 4 x osx +o Is WA. Let U> coex dus —awndx case S* f sin"udu and fwstudu whee Ais on een wnkgev" aa) Ssin*xax Sela: Ssin *OK = —_— I= a a = fae — J} 52% dx = oSdx — Af msrxdx = 4x ysinex AG V bet = 2x du. = Ww Case4® fsa cos" x dx , where oh m ard n ave even wm) S s0"xoostxdx . Solin Pants ook dA = gu anton dn = lle = ise 4 Jussndx + tS werd subserlbe,0n yaubuloe, Be @enginerdmnath - 4 ~ me + 4 See dx sf ra muds wv in uly | “hae W\sq « phe whegar We we yay = fan! y kana cok4= ook™%4 Cost) = tan ™u (sect ut) = ok” “a(oscru 1) al) J tan’xdx Sole: tana = Plane sectn-Ddoe = JS tarmysecdtxdas ~ Seana lebus ture J udu — Stanxay dus sexdx ME |n Yor) ) : ints + Wl ox | EC B) Jak! ond Sorin: Soottardx = Sok 9x (050 AOA = J cok xecaxdx ~S oad = ete eck? 9x) — — SEse3x -1) dx § wb 3x +L kK REC uw | dows Psecrudy or [b%"udy whee n 15a posite eon Inkyor We witle seclus sec™*u secu eschus csc U ecru Sesct = Carta Hy secu = bot*u 4) me esc csckad $3) x . > Subsorilbe, 0 yoatulbe, 20> Sovn Poses = JUttx +1 * esc xdx Geng\nerdinat = J ot'x coctxdx + 2(cat’xcsctxdx + Ses?xdy = wt? ~% cot — atat+e ose! Precrudu 0 fosctudy ube nb a posthve ud inkegen> To inkegrale odd prot of secant and coscant we we yokyation by puts 34) Ssxixdx Qyq: bet y= Sean dv2 Sectxdx du> secrtanndX Y= tax Socckd x= seextanx — SJ seem KanKax = secatany, — Ssecx (se@x- Ax Sedade > seextanx ~ Jecx + Sseuxdx dsccradye = seer tanx + \n) sen + tanx | Seedxan= | (seastame + nl oe t+ tan] ) $C cage Stan" secudy of Paty csc'udy where Vis a psite een inte gen, 35) J fan? sec’ndx Solin: J fan8x seetnace = S Yat x( tana 1) sein yaus tone = Sta xs00% dm + Pravtnsecnde dus secydx = + tan X + thntate wR Case: SYonuseo™udu of Scot™u cso "udy jubete MIs a positive, odd Weyer a) Start % SEC MAK hus scx Sova Sax sed ydor = J tanix sectx secatunndX — dx coxtancdy =! (8 1)" seobg (seem tan dx) Se sec’ (eextaiixdx) -z. sack (oeaMtanx an) + Psecbs (seextaneax) =| cece - 2 segly + Ls0ec? 7% +] subscrive-on vypatuloe Po 4 7 @ engunerdinath Case? Ptonuseo™udy of Scok"u cso "udy jubete MIs a postive. un indegey and miso positive odd \nleger ‘the inkegrand can ve. expessed in ms of WH youre oF secant of coca SMart seconds =P oe) se xdx = JoecBxdx ~Psec%dx > pees In OES Integration By Trigonomehic: Substihition, Case the integrand contain, on expresamry wf the ym aur ,a>0 let ug asind Ht al 4x J a “ SYn. leh X= 3snB , dx= 3as0d0 Nox? = V9= Got = 3A GST = 2eosd je. S228 costda) = Sett¥as = Secs “ae =-at) -B +c 3 x Banu sm p= and “prs 0 =) kK, B= sit? - To find cat B , refordo Ryire on lett yuo wtp < 12Ex™ subsite 9 youtube, Rf x Genginerdinath AQ=-x% x there fie: aa = Se - aie ~ sy te coger a The ankegrand wntams an) exgrecson uf the fim ATaHUF aso let u= atand 28) | VS ax Soym: ket A= VT tanY das AT seco AWTS = NStarte to = VS 1500 = V5 od Sweex = Semen (E29 dO) = BY sec?d dd j Use the resulkin Problem 34: =F secdtan® +S ln]sece + tané | tc Fon the ques, tan8 = 9B ) 8x8 - Ab ein = BBE Hes T Er htc 2d KTS + E In| dere tx) Bye tc [Lore + F In| ere +x] + vee 0, = --S ne FC 24 Coss The wnleqrard Colains an expesaon of He Arm Muar ,a>0 tel u= asece ssubsoite 9 yale, LP 34) JS NA @enginerdmatn Sy lee X= Bsec8 | AX~ Ssecotand ad Nea = Namco a > SNe ~ Stand a sedi d= tL Dene : aS 34nd 3 J evs ade -id Cit @s20) do =tet¢bsn20) +¢ a7 7) (8 + sindcos®) +o Fim the Rgure: sind = oe ) asd = S- So¥n De vompace inho san of partial Factions Xe! = A 4 8B 4 ¢ XOF2) OF/) x X2 nt | ao + Subsofilte 09 utube, Af shy Be “i me “ 4 2 Genguyordmath wt. 2, ey tS KOM K X-2 xF1 PH ye Lp yd pdx — 2p Seah 3 x pees ao [2 |ni) +b Inpee} - Slob + 2 Bt 4) J Ka-T oe Sol’: Devorpose nhs sam of partial Pactions : Fl 2 AY ay c + + tx Gp * Gay eT Sohiy fy A,B, C,D, and E x) 2. et = 3 KOPP? ar + = = +—_le fay oF 2 P wel o ne US ere we alle apt wep aye res = Ss _ ar ib s\n] Bep ~ Gore) Fo bbe aite _ |e + Bx-4 og x ahaa Pe Mee] te Sy Jeb X= BY dys bFAy Plath) _ [2 Sg dae SAP JS ue Subserilve-0n youbuloe, £ fb Dive at fist , we have: Beunerinath Jp xh _ a Te \ ea ax (2 Bye 1 + Sh ) de 2 6(52?~ da dy -2 + tee) te [fy bb + G x bax ba + bran W+C) J dx 9) Us 4 wx \ 2d% Son: bet B= tan gx ax= 422 \-2 2e COSK= ' swxX = —— ee 7 I+ 22 ax . 2dy FS J [tae |- 2 _ + ERE re ee = a * (1427) ~ 24 + (-2%) de =-| Cc Ae = = n\bo\ + EI n|i-tangxl tc) u APPLICATIONS OF INTEGRAL CALCULUS D\ py es 4) Find the ate of He region bounded y canes Y= x" ond y= ~-W+4y- Sov: lek pg) = ae ba 1 9%) =x. Get Hee point (6) of \nterge ction 4 Fix) = goo s 0 44K = x Gan Jame = 48) = es aru d subsoil on ype, RP ian @ arnatn axUr2) =O x eens K=O and yo 7 day vertical) spe me 4 he J Cap Ye Vb . f° [o-* 4x) — x* J dx = Sy D4 KB > - 2x4 an |* --3o° + 205) - F 305128) = -H4y-(oto st 45) Find the yolume of the solid generaled by rrotngg about he natnls ‘he veqion ouvded wy the paral y=x*+1 avd the hme yout, Vy: SO Co the powntis) of inlersection , of tw cates tek POD = xt3 and glk) = #4) y upper = Plad= WB Foy = 91x) AEB = XE) quepyekt LAY e) W-xX-2 =O (AIM -2) <0 Re-| X= 2 % Re 1. sce on yb 9 Us York ship fir dick eto: asia me 7h, u L V2) [Yom ~ Yor) a “7 Cw Craiydda 1 oof (=x! - ¥ 46x45) ax L

You might also like