Integral Calculus What is integral calculus?
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1 Intgral Calulus What is intgral alulus? In diffrntial alulus w diffrntiat a funtion to obtain anothr funtion alld drivativ. Intgral alulus is onrnd with th opposit pross. Rvrsing th pross of diffrntiation and finding th original funtion from th drivativ is alld intgration or anti-diffrntiation.
2 Th Indfinit Intgral If F is a funtion of and its drivativ, d[ F ] F f d thn F is alld an indfinit intgral or simply an intgral of f. Symbolially this is writtn as, f d F
3 Th symbol is th intgral sign, f is intgrand ^wkql,hh& and is th onstant of intgration. Constant may hav diffrnt valus and aordingly w an hav diffrnt mmbrs of fd family. For ampl, if y =, th drivativ of y is dy d Whn dy is givn, to find y w hav to follow th d opposit pross of diffrntiation. Thus, d
4 Howvr, dy d diffrntiabl funtions of y. may b th drivativ of diffrnt For ampl it is quivalnt to th drivativs of y = +, y = - 5, y = + a t. Thus, if w add a onstant to th intgral it will math with th diffrntial funtion. d
5 Ruls for Indfinit Intgration Rul : th Powr Rul n d n n ; n g. d 6
6 Rul. th Intgral of a Multipl kf d k f d ; k is a onstant g. d d Rul.th Intgral of a Sum or Diffrn f g d f d g d 5 d d 5 d 5 d
7 Rul. Gnralizd Powr Funtion Rul a n b a d b a n n d 9 d / / / 6 g. i. g. ii.
8 Rul 5. th Logarithmi Rul d ln 0 i. d d ln ii. d d ln
9 Rul 5. If f is a funtion of and its diffrntial offiint with rspt to is f. Th drivativ of ln[±f] is d ln[ f ] f d f f d ln f f
10 g. i. ln d g. ii. d Th drivativ of th dnominator is, and to apply th abov rul th numrator should b multiplid by as suh dnominator, too. d d ln
11 Rul 6: th Eponntial Rul d d d d d g. i. i.
12 f d f f ii. g. i. d iii. ] [, f d d y Whn f f f d f f f
13 g. i. Find indfinit f = and intgral f of d To apply th abov rul w hav to adjust th original funtion as, Thr is not a fundamntal d diffrn btwn th original funtion and this funtion. W an apply th abov rul for this funtion d d
14 Rul 8: th Substitution rul d Lt u = +; thn du/d = or d = du/. Now du/ an b substitutd for d of th abov funtion. d u u du udu
15 g. 8 d If w dfind u = + 8 and du/d =. From this du d 8 d u u du d u u du 8
16 Th Dfinit Intgral Th indfinit intgral of a ontinus funtion f is: f d F If w hoos two valus of in th domain, say a and b b > a, substitut thm sussivly into th right sid of th abov quation and form th diffrn w gt a numrial valu that is indpndnt of th onstant. F b F a F b F a This valu is alld th dfinit intgral of f from a to b. a and b ar lowr and uppr limits of intgration, rsptivly.
17 Evaluat th following dfinit intgrals 56. d d a F b F a F b F F d f b a b a Now, w will modify th intgration sign to indiat th dfinit intgral of f from a to b as:
18 b a a b b a k k d k d
19 Th Dfinit Intgral as an Ara Th ara of th rgion boundd by th urv y = f, and by th ais, on th lft by = a, and on th right by = b is givn by, Y A y=f Ara A f d b a 0 a b X If th urv y = f lis blow th ais, thn Ara b a A f d
20 Y y = f A 0 a b B X b Th ara A + B = f d f d a b
21 IF f and g ar th two funtions of and f > g Y y = f J A B y = g 0 a Ara A = Ara J Ara B b a b f d g d b a X
22 g.. Dtrmin th ara undr th urv givn by th funtion y = 0 ovr th intrval 0 to 5. Y 0 A y = 0 Y Ara A = 0 0 d
23 Eg. Find th ara boundd by th funtions y =, and y = 0 and Y ais. y = A = 0 0 d d 0 0 A y =
24 . Dtrmin th ara btwn th urv of f = 0 - X and th X ais for valus of X = to X = 7. f 0 a AREA = ab + bd b d 0 X f = 0 X
25 AREA ab = 5 0 X dx [0X X ] 5 AREA bd = X dx [0X - X ] 7 5 AREA = ab + bd = 8
26 Eonomi Appliations. If th marginal ost MC funtion of a firm is C and th fid ost C F = 90. Find th total ost funtion TC.. If th marginal ost MC funtion of a firm is MC q 6q q q 6 0.Q and fid ost of th funtion is 088. Find th total ost funtion TC.. If th marginal saving funtion of a ountry is S Y 0. 0.Y. If th aggrgat saving S is zro whn inom Y is 8. Find th saving funtion SY.
27 . Consumr s dmand funtion for a givn ommodity has bn stimatd to b P = 0 Q whr, P is th pri of a unit of th ommodity and Q is th pr apita onsumption of th ommodity pr prson pr month. Dtrmin a th total pnditur and b th onsumr surplus whn th pri of a unit is 5.
28 Dtrmin th produr surplus paid to th nurs if th prvailing wag rat is 9 pr hour. 5. If th supply funtion of a ommodity is P = Q whr, P is th pri pr unit and Q is th numbr of units sold ah day. Find th produr surplus whn th pri of a unit of th ommodity is If th willingnss of a nurs to provid hr srvi is dfind by th supply funtion W = H whr, W is th wag rat pr unit H is hours of work providd ah wk.
29 9. Suppos that t yars from now, on invstmnt will b gnrating profit at th rat of ' P t 50 t hundrd dollars pr yar, whil a sond invstmnt will b gnrating profit at th rat of ' P t 00 5t hundrd dollars pr yar. Pt and Pt, satisfy P t P t for th first N yars 0 t N. a For how many yars dos th rat of profitability of th sond invstmnt d that of th first? b Comput th nt ss profit for th tim priod dtrmind in part a. Intrprt th nt ss profit as an ara.
30 0. Suppos that whn it is t yars old, a partiular industrial mahin gnrats rvnu at th rat R t = 5,000-0t dollars pr yar and that oprating and srviing osts rlatd to th mahin aumulat at th rat C t, t dollars pr yar a How many yars pass bfor th profitability of th mahin bgins to dlin? b Comput th nt arnings gnratd by th mahin ovr th tim priod dtrmind in part a
31 ' tla;rd NdKavhla i yd jq mßfndal b,a ï Y%s;h Dq = 5-q fõ' i. NdKavfhka tall la ñ,g.ksu i yd mßfndalhd f.ùug iqodkï uqo,a m%udkh fldmuk o@. fudag¾ r: ghr ksiamdok iud.ula wia;fïka;= lr w;s wdldrh wkqj.kqïlrjka úiska tallhl ñ, rmsh,a Dq= -0.q + 90 jk úg ghr q ^oyia& m%udkhla b,a ï lrk w;r wdh;khtu m%udkh imhkafka Sq = 0.q + q ñ,g h' i. iu;=,s; ñ, yd m%udkh.kkh lrkak' ii. iu;=,s;fhaos mßfndal w;sßla;h yd ksiamdol w;sßla;h.kkh lrkak'
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