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1 Integrl Clculus This unit is esigne to introuce the lerners to the sic concepts ssocite with Integrl Clculus. Integrl clculus cn e clssifie n iscusse into two thres. One is Inefinite Integrl n the other one is Definite Integrl. The lerners will lern out inefinite integrl, methos of integrtion, efinite integrl n ppliction of integrl clculus in usiness n economics.

2 School of Business Blnk Pge Unit- Pge-8

3 Bnglesh Open University Lesson-: Inefinite Integrl After completing this lesson, you shoul e le to: Descrie the concept of integrtion; Determine the inefinite integrl of given function. Introuction The process of ifferentition is use for fining erivtives n ifferentils of functions. On the other hn, the process of integrtion is use (i) for fining the limit of the sum of n infinite numer of infinitesimls tht re in the ifferentil form f ( ) (ii) for fining functions whose erivtives or ifferentils re given, i.e., for fining nti-erivtives. Thus, reversing the process of ifferentition n fining the originl function from the erivtive is clle integrtion or nti-ifferentition. The integrl clculus is use to fin the res, proilities n to solve equtions involving erivtives. Integrtion is lso use to etermine function whose rte of chnge is known. In integrtion whether the oject e summtion or nti-ifferentition, the sign, n elongte S, the first letter of the wor sum is most generlly use to inicte the process of the summtion or integrtion. Therefore, f ( ) is re the integrl of f () with respect to. Agin, integrtion is efine s the inverse process of ifferentition. Thus if g( ) = f ( ) then f ( ) = g( ) + c where c is clle the constnt of integrtion. Of course c coul hve ny vlue n thus integrl of function is not unique! But we coul sy one thing here tht ny two integrls of the sme function iffer y constnt. Since c coul lso hve the vlue zero, g () is one of the vlues of f ( ). As c is unknown n inefinite, hence it is lso referre to s Inefinite Integrl. Some Properties of Integrtion The following two rules re useful in reucing ifferentile epressions to stnr forms. (i) The integrl of ny lgeric sum of ifferentil epression equls the lgeric sum of the integrls of these epressions tken seprtely. i.e. [ ( ) g( )] = f ( ) ± f ± g( ) (ii) A constnt multiplictive term my e written either efore or fter the integrl sign. i.e. cf ( ) = c f ( ) ; where c is constnt. Reversing the process of ifferentition n fining the originl function from the erivtive is clle integrtion. f ( ) is re the integrl of f () with respect to. Business Mthemtics Pge-9

4 School of Business Some Stnr Results of integrtion A list of some stnr results y using the erivtive of some wellknown functions is given elow: ( i ) = + c ( ) = ( ii) n+ n = + c n + ( iii ) = log + c ( iv) e = e + c n+ = n + n, (log ) = ( ) e = e n ( v) = + c ( ) = log log ( vi ) sin = cos + c ( cos ) = sin ( vii ) cos = sin + c (sin ) = cos ( viii ) sec = tn + c (tn ) = sec ( i ) cosec = cot + c ( cot ) = cosec ( ) sec tn = sec + c (sec ) = sec tn ( i ) cosec cot = cos ec + c ( ii ) ( ii ) ( iii ) ( cosec) = cose cot - = sin + c (sin ) = = tn + c (tn ) = + + = sec - + c (sec ) = + ( iv ) tn = log sec + c (log sin ) = cot ( v ) cot = log sin + c (log sec ) = tn Unit- Pge-

5 Bnglesh Open University Illustrtive Emples: Emple -: Evlute = + c Emple-: Evlute = + c Emple-: Evlute = + c Emple-: Evlute ( + ) + ) ( = Emple-5: + = + + c Evlute ( 7 + 6) ( 7 + 6) Emple-6: Evlute = = c = Business Mthemtics Pge-

6 School of Business = c Emple-7: Evlute + + ( + ) = + + ( + + )( + ) ( + ) = = + + = ( + ) ( ) + c 6 6 Emple-8: Evlute ( 5e + ) ; (5e + ) = e 5 + = 5e + + log + c Unit- Pge-

7 Questions for Review: Bnglesh Open University These questions re esigne to help you ssess how fr you hve unerstoo n cn pply the lerning you hve ccomplishe y nswering (in written form) the following questions:. Integrte the following functions w. r. t. 6 (i) ( 5 + ) (ii) (iii) ( ) (iv) ( + ) (v) + + (vi) ( e ) Business Mthemtics Pge-

8 School of Business Certin functions cn e integrte quite simply y pplying rules of integrtion. Sustituting new suitle vrile for the given inepenent vrile n integrting with respect to the sustitute vrile cn often fcilitte Integrtion. Lesson-: Methos of Integrtion After stuying this lesson, you shoul e le to: Descrie the methos of integrtion; Determine the integrl of ny function. Introuction In compring integrl n ifferentil clculus, most of the mthemticins woul gree tht the integrtion of functions is more complicte process thn the ifferentition of functions. Functions cn e ifferentite through ppliction of numer of reltively strightforwr rules. This is not true in etermining the integrls of functions. Integrtion is much less strightforwr n often requires consierle ingenuity. Certin functions cn e integrte quite simply y pplying rules of integrtion. A nturl question is tht wht hppens when rules of integrtion cnnot e pplie irectly. Such functions require more complicte techniques. This lesson iscusses four techniques tht cn e employe when the other rules o not pply n when the structure of the integrn is of n pproprite form. In generl, eperience is the est guie for suggesting the quickest n simplest metho for integrting ny given function. Methos of Integrtion The following re the four principl methos of integrtion: (i) Integrtion y sustitution; (ii) Integrtion y prts; (iii) Integrtion y successive reuctions; (iv) Integrtion y prtil frction. Integrtion y Sustitution Sustituting new suitle vrile for the given inepenent vrile n integrting with respect to the sustitute vrile cn often fcilitte Integrtion. Eperience is the est guie s to wht sustitution is likely to trnsform the given epression into nother tht is more reily integrle. In fct this is one only for convenience. The following emples will mke the process cler. Emple-: 5 Evlute ( + ) Let, + = z = z = z 5 5 ( + ) = z. z Unit- Pge-

9 Bnglesh Open University = z z 5 6 z =. + c 6 6 ( + ) =. + c 6 Emple-: 5 Evlute ( + ) Let, + = z = z = z 5 ( + ) = Emple-: Evlute z 5. z = z 5 z 6 z =. + c 6 6 = ( + ) + c e Let, = z = z = z e = e z. z = e z + c = e + c Emple-: Evlute + Let, + = z = z. Business Mthemtics Pge-5

10 School of Business Integrtion y prts is specil metho tht cn e pplie in fining the integrls of prouct of two integrle functions. = z + = z. z = z z z =. + c = ( + ) + c Integrtion y Prts Integrtion y prts is specil metho tht cn e pplie in fining the integrls of prouct of two integrle functions. This metho of integrtion is erive from the rule of ifferentition of prouct of two functions. If u n v re two functions of then, ( uv) = v u + u v u v v u = ( uv) Integrting oth sies with respect to, we get u u v v Putting u = f(), = ( uv) v u = uv v v u = g() then v = g() ( ) g( ) f ( ) g( ) [ f ( ) f = g( ) ] Thus integrl of the prouct of two functions = st function integrl of the n integrl of (ifferentil of st integrl of n ). It is cler from the formul tht it is helpful only when we know integrl of t lest one of the two given functions. The following emples will illustrte how to pply this rule. Emple-5: Evlute e Unit- Pge-6

11 Bnglesh Open University e = e { ( ) e } Emple-6: Evlute log log = log.. Emple 7: Evlute = e e = e e + c = log. { (log ). } = log.. = log = log + c. log log = log { (log ) } = log... = log = log + c 9 Integrtion y Successive Reuction Any formul epressing given integrl in terms of nother tht is simpler thn it, is clle reuction formul for the given integrl. In prctice, however, the reuction formul for given integrl mens tht the integrl elongs to clss of integrls such tht it cn e epresse in terms of one or more integrls or lower orers elonging to the sme clss; y successive ppliction of the formul, we rrive t integrls which cn e esily integrte n hence the given integrl cn e evlute. Emple-8: Evlute e Any formul epressing given integrl in terms of nother tht is simpler thn it, is clle reuction formul for the given integrl. Business Mthemtics Pge-7

12 School of Business e = e { ( ) e } e = e e e = e e e e = + e e e = + e e + c 9 7 Mny rtionl functions eist which cnnot e integrte y the rules of integrtion. Integrtion y Prtil Frction Rtionl functions hve the form of quotient of two polynomils. Mny rtionl functions eist which cnnot e integrte y the rules of integrtion presente erlier. When these occur, one possiility is tht the rtionl function cn e restte in n equivlent form consisting of more elementry functions n then ech of the component frctions cn e esily integrte seprtely. The following emples illustrte the ecomposition of rtionl function into equivlent prtil frctions. Emple-9: Evlute = + + ( + )( + ) = + + = log (+) log (+) + c Emple-: + Evlute + + = ) + ( + = + = log (+) + c Unit- Pge-8

13 Bnglesh Open University Emple-: Evlute = ( + ) 5 = - + ( + ) = 5 log( + ) + + c + Emple : Evlute = ( + ) = ( + ) = + log( ) + c Emple : Evlute + = = + + = log + + log( + ) + c Business Mthemtics Pge-9

14 School of Business Questions for Review These questions re esigne to help you ssess how fr you hve unerstoo n cn pply the lerning you hve ccomplishe y nswering (in written form) the following questions:. Integrte the following functions w. r. t. (i) ( + ) 8 (ii) ( + ) (iii) log (iv) e (v) ( + )( + ) Unit- Pge-

15 Lesson-: Definite Integrl After stuying this lesson, you shoul e le to: Descrie the concept of efinite integrl; Evlute efinite integrls. Introuction Bnglesh Open University In Geometry n other ppliction res of integrl clculus, it ecomes necessry to fin the ifference in the vlues of n integrl f() for two ssigne vlues of the inepenent vrile, sy,,, ( < ), where n re two rel numers. The ifference is clle the efinite integrl of f() over the omin (, ) n is enote y f ( )...( i) If g() is n integrl of f(), then we cn write, [ g( ) ] = g( ) g( )...( ) f ( ) = ii Here f ( ) is clle the efinite integrl, s the constnt of integrtion oes not pper in it. If we consier [g() + c] inste of g(), we hve [ g( ) + c] = g( ) + c g( ) c = g( ) g( )...( ) f ( ) = iii Thus, from (ii) or (iii), we get specific numericl vlue, free of the vrile s well s the ritrry constnt c. This vlue is clle the efinite integrl of f() from to. We refer to s the lower limit of integrtion n to s the upper limit of integrtion. Properties of Definite Integrl Some importnt properties of efinite integrl re given elow: ( i ) f ( ) = f ( z) z ( ii ) f ( ) = f ( ) ( iii) f ( ) = c f ( ) = f ( ) + ( iv ) f ( ) ( v ) cf ( ) = c f ( c ) Business Mthemtics Pge-

16 School of Business ( vi) n ( vii ) f ( ) = f ( f ( ) = n (viii) (i) + f ( ) = f ( ) = f ( f ( f ( ) ) ) ) Illustrtive Emples: Emple-: Evlute c c = c Emple-: Evlute = Emple-: 9 Evlute 9 if = c [ ] = c ( ) = 9 - = = if f( + ) = f( - ) = 9 = f() f() Emple-: Evlute ( - + 5) ( - + 5) = + 5 = 7 Unit- Pge-

17 Bnglesh Open University Emple-5: Evlute Let = z = z when =, then z = when =, then z = 5 Emple-6: 5 = z = [ ] 5 z Evlute + 9 Let + 9 = z = z when =, then z = 9 when =, then z = 5 log z + 9 = z z = Emple-7: π 5 Evlute + sin π 9 Let I = + sin π π = π = (log 5 log ) z = ( π ) = + sin( π π ) π + sin + sin π I = π = + sin sin cos π sec π tn sec = [ tn sec ] π = π I = π. π π π = π (π ) = + sin Business Mthemtics Pge-

18 School of Business An integrl is si to e proper integrl when it is oune n the rnge of integrtion is finite. When the limit eists, the integrl is si to e convergent to tht limit n when the limit oes not eist, the integrl is si to e ivergent to tht limit. Proper n Improper Integrls Proper Integrls: An integrl is si to e proper integrl when it is oune n the rnge of integrtion is finite. For emple, f ( ), f ( ) etc. Improper Integrls: When the rnge of integrtion is finite ut the integrn is unoune for some vlues in the rnge of integrtion, then it is clle the improper integrl of first kin. e.g. ( )( etc. ) When the rnge of integrtion is infinite ut the integrn is oun, then it is clle improper integrls of secon kin. e.g., f ( f ( ) etc. These types of improper integrls re etermine s if w ( ) = lim w ) f f ( ).,, f ( ), When the limit eists, the integrl is si to e convergent to tht limit n when the limit oes not eist, the integrl is si to e ivergent to tht limit. If, however, the limit oes not converge or iverge then it is si to e oscilltory. Emple-8: Determine whether the improper integrl ivergent. e is convergent or e e = lim = lim ( e ) = Thus the given improper integrl is ivergent. Emple-9: Determine whether the improper integrl ivergent. e is convergent or Unit- Pge-

19 e = lim e = lim ( e e ) = lim ( e ) = = Bnglesh Open University Thus the given improper integrl is convergent n its vlue is. Multiple Integrls Integrtion of function in one vrile genertes n re ( surfce) from line. A function in two vriles genertes volume from surfce. Becuse function in two vriles escries surfce with ifferent curvture in ech irection, oth vriles re responsile for generting the pproprite volume. To fin the volume, the integrl must e tken with respect to oth vriles. Emple-: Fin the vlue of + ( + )y ( )y = [ ( + )y ] A function in two vriles genertes volume from surfce. = 5 Emple-: Fin the vlue of = [ y + y ] = [ + ] = [ + ] 5 yzy yzy = [ 5 5 yz ]y Business Mthemtics Pge-5

20 School of Business = [ = [ = yz ] 5 y [ =.. = 5 y y ] y ] Unit- Pge-6

21 Questions for Review: Bnglesh Open University These questions re esigne to help you ssess how fr you hve unerstoo n cn pply the lerning you hve ccomplishe y nswering (in written form) the following questions:. Evlute the following integrls: (i) e (ii) (+ log (iii) (iv) π + (v) log sin ) (vi) (vii) ( ( + y + y ) y + z ) yz Business Mthemtics Pge-7

22 School of Business Lesson-: Applictions of Integrtion in Business After stuying this lesson, you shoul e le to: Epress the importnce of integrtion in Business n Economics; Apply integrtion in ifferent types of usiness ecisions. Mrginl cost is the chnge in totl cost from n incrementl chnge in output n only vrile costs chnge with the level of output. Introuction The knowlege of integrtion is wiely use in usiness n economics. For emple, net investment I is efine s the rte of chnge in cpitl stock formtion K over time t. If the process of cpitl formtion is K( t) continuous over time, I(t) = = K ( t). From the rte of t investment, the level of cpitl stock cn e estimte. Cpitl stock is the integrl with respect to time of net investment. Similrly the integrl cn e use to estimte totl cost from the mrginl cost. Since mrginl cost is the chnge in totl cost from n incrementl chnge in output n only vrile costs chnge with the level of output. Economic nlysis tht trces the time pth of vriles or ttempts to etermine whether vriles will converge towrs equilirium over time is clle ynmics. Thus, we cn use integrtion in mny usiness ecision mking processes. In this lesson, we iscuss out few smple pplictions of integrtion. The following emples illustrte smple pplictions of integrl clculus. Illustrtive Emples: Emple-: Fin the re oune y the curve = n =. We know tht, Are = Emple-: y = y =, the -is n the lines = = The mrginl cost function of prouct is given y C = q +.q, where q is the output. Fin the totl n q verge cost functions of the firm ssuming tht its fie cost is $5. C Given tht = q +.q q Integrting this with respect to q, we get, C = ( q +.q ) q. C = q 5q + q + K 6 Unit- Pge-8

23 Bnglesh Open University Now the fie cost is 5; i.e., when q =, C = 5. K = 5. Hence the totl cost function is, C = q 5q + q + 5 An the verge cost is C. = 5q + q 5 + q q Emple-: The mrginl revenue function. The revenue function is R ( ) = R ( ) = ( = K.. Fin the revenue R ( ) = ) We know tht when no prouct is sol then the revenue is zero. i.e., when =, R =. Thus, K =. Thus the revenue function is Emple-: R ( ) + = 5 +. The mrginl cost function for certin commoity is MC = q q + 5. Fin the cost of proucing the th through the 5 th units, inclusive. 5 (q 5 q + 5)q = (q q + 5)q q q + 5q = 5 = [ ] 5 Emple-5: A Compny etermines tht the mrginl cost of proucing units of prticulr commoity uring one-y opertion is MC = 6 59, where the prouction cost is in ollr. The selling price of commoity is fie t $9 per unit n the fie cost is $8 per y. (i) Fin the cost function. (ii) Fin the revenue function. (iii) Fin the profit function. (iv) Fin the mimum profit tht cn e otine in one-y opertion. Business Mthemtics Pge-9

24 School of Business Given tht MC = 6 59 FC = 8 P = 9 (i) Cost function, TC = MC (ii) Revenue = P = 9 = ( 6 59) = c = (iii) Profit = TR TC = 9 ( ) = (iv) Profit y = y = 6 6 y For mimum or minimum, = 6 6 = = y Agin, = 6 Hence the require profit mimizing sles volume is =. Require mimum profit y = = 6() 8() 8 = $78. Emple-6: After n vertising cmpign prouct hs sles rte f(t) given.5t y f ( t) = e where t is the numer of months since the close of the cmpign. (i) Fin the totl cumultive sles fter months. (ii) Fin the sles uring the fourth month. (iii) Fin the totl sle s result of cmpign. Let F(t) is the totl sle fter t months since the close of the cmpign. The sle rte is f(t). t F(t) = f ( t) t (i) The totl cumultive sles fter months, F() = f ( t) t =.5t e t f ( t) t Unit- Pge-5

25 Bnglesh Open University.5 =.5t [ e ] = [e.5 ] = (. ) = 55 units..5t (ii) Sles uring the fourth month = e t.5 e t.5t e.5 t = [ ] = [e. e.5 ] = (.5 ) = 75.6 units. (iii) Totl sles s result of cmpign =.5t e t.5 e t t e.5 t.5 = [ ] = ( ) = units. Emple-7: If $5 is eposite ech yer in sving ccount pys 5.5 % per nnum compoune continuously, how much is in the ccount fter yers? Given tht, pyment per yer, P = 5 Rte of interest, r =.55 Time, t = t Amount of the future vlue A = Emple-8: = t P e r t.55t 5e t 5 t = [ e ] = 99.9 [e. e ] = 99.9(.676) = $7. If the mrginl revenue n the mrginl cost for n output of commoity re given s MR = 5 + n MC = +, n if the fie cost is zero, fin the profit function n the profit when the output =. Business Mthemtics Pge-5

26 School of Business Given tht, MR = 5 + MC = + Profit = TR TC = MR MC = (5 + ) ( + ) = c ) ( + + c ) ( Since the fie cost is zero so tht c = ; for =, totl revenue = Profit = + When =, the profit = ( ) () + () =. Emple-9: Fin the totl cost function if it is known tht the cost of zero output is c n tht mrginl cost of output is We re given tht, Mrginl cost (MC) = TC = + TC = ( + ) TC = When =, TC = c, so, K = c. + K + Hence the totl cost function is given y, TC = Emple-: + c + Let the rte of net investment is given y I ( t) = 9t, fin the level of cpitl formtion in (i) 8 yers (ii) for the fifth through the eighth yers. 8 (i) K = t 9 t = 6t = (ii) K = 9 t t = 6t = Unit- Pge-5

27 Questions for Review: Bnglesh Open University These questions re esigne to help you ssess how fr you hve unerstoo n cn pply the lerning you hve ccomplishe y nswering (in written form) the following questions:. Mrginl cost is given y MC = 5 + Q 9Q. Fie cost is 55. Fin the (i) totl cost (ii) verge cost, n (iii) vrile cost functions.. Mrginl revenue is given y MR = 6 Q Q. Fin the totl revenue function n the emn function. 5. The rte of net investment is I = t 75. Fin the cpitl function K. n cpitl stock t t = is Business Mthemtics Pge-5