The Evaluation Theorem


 Magdalene Simon
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1 These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for inclss presenttion nd should not e regrded s sustitute for thoroughly reding the textook itself nd working through the exercises therein The Evlution Theorem As we hve lerned y experience, it is very difficult nd tedious to compute definite integrls, fxdx, using the definition of the definite integrl, fxdx lim n R n In fct, it is usully not possile to do this unless the Riemnn Sum, R n, tht is involved contins specil summtion formuls such s n i i nn 2 tht llow us to simplify R n in such wy tht lim n R n cn e computed Some previous exmples tht we studied in which we were le to compute definite integrls using the definition were x 2 dx 0 3 nd 2 x 3 dx 375 Some other exmples in which we were le to compute the exct vlue of definite integrl y using sic geometry were 5 32t 48dt 60 0 nd x 2 dx 2 The theorem tht we re now out to stte, which the uthor or our textook, Jmes Stewrt, clls The Evlution Theorem, gives us considerly esier wy to evlute mny definite integrls Theorem (The Evlution Theorem) Suppose tht the function f is continuous on the intervl, nd suppose tht F is n ntiderivtive of f on, Then
2 fxdx F F Exmple Use the Evlution Theorem to find the vlues of x 2 dx, 0 2 x 3 dx, t 48dt, nd 0 sinxdx 2
3 Remrk It is not lwys esy to use the Evlution Theorem Sometimes it cn e very difficult to find n ntiderivtive of the integrnd, f In fct, sometimes it is impossile! An exmple of difficult (ut not impossile) prolem on which the Evlution Theorem cn e used is x 2 dx An exmple of prolem on which it is impossile to use the Evlution Theorem is 0 sinx 2 dx 3
4 Why Does the Evlution Theorem Work? Suppose tht f is continuous on the intervl, nd suppose tht F is n ntiderivtive of f For ny given positive integer, n, we use the usul method of sudividing the intervl, into n pieces: x n n x 0 x x n x 2 2 x n x n n x n x n n x n nd, s usul, we define n R n fx i x n i Now, oserve tht F F Fx n Fx n Fx n Fx n2 Fx 2 Fx Fx Fx 0 n Fx i Fx i i Since the function F is differentile on ech of the suintervls x i,x i, then y the Men Vlue Theorem, there exists t lest one point x i x i,x i such tht F x i Fx i Fx i x i x i Fx i Fx i x n From this, we conclude tht Fx i Fx i F x i x n However, recll tht F is n ntiderivtive of f (which mens tht F f) Therefore, Fx i Fx i fx i x n We thus see tht n F F fx i x n i Tking the limit s n on oth sides of the ove eqution gives us 4
5 F F fxdx 5
6 Indefinite Integrls The Evlution Theorem shows tht there is close connection etween the processes of integrtion nd ntidifferentition For this reson, we sometimes use the integrtion symol,, s symol for ntidifferentition We write fxdx to denote the definite integrl of f over the intervl,, nd we write fxdx to denote n indefinite integrl of f, mening ny ntiderivtive of f Thus, for exmple, we could write x 2 dx 3 x3 C nd e x dx e x C 6
7 Exmple Find the following indefinite integrls: x n dx (ssuming tht n is constnt with n ) x dx sinxdx cosxdx sec 2 xdx e x dx dx Remrk The notion of n indefinite integrl s defined in most textooks (including ours) is not techniclly correct The sttement tht F is n ntiderivtive of f does not mke sense unless we refer to some intervl Thus, sttement tht does mke sense is F is n ntiderivtive of f on the intervl, A more correct nottion for n indefinite integrl would mke reference to the intervl, Thus, more correct nottion for n indefinite integrl might look something like, In most situtions encountered in elementry clculus, this nunce does not relly mtter (which is proly why most textook uthors ignore it) For exmple, the sttements, x 2 dx 3 x3 C nd, cosxdx sinx C re correct no mtter wht intervl, we use However, the sttement, x dx lnx C might or might not e correct, depending on wht the intervl, is If, is n intervl of positive numers, then the sttement is correct, ut if, is n intervl of negtive numers, then the sttement is not correct 7
8 How To Mke Up n Integrtion Prolem Tht You Cn Do But Noody Else Cn (Unless They Are Very Clever) Wnt to impress you friends? The Evlution Theorem tells us tht if F is n ntiderivtive of f, then fxdx F F Therefore, the definite integrl of f over the intervl, is esy to compute if you know wht F is One wy to mke up hrd prolem (tht you cn esily do ut others will hve troule with) is to strt with some very complicted function F (tht you mke up), then compute its derivtive, f Chnces re tht if F is very complicted looking, then f will lso e very complicted looking Then sk your friends to compute fxdx You will know how to compute this integrl ecuse you know wht F is However, your friends might hve hrd time Here is n exmple of this: Let s tke The derivtive of this function is Fx x 2 2x 3 x 2 3 8
9 F x x2 3 d dx x 2 2x 3 2 x 2 2x 3 2 d dx x2 3 x x2 3 2 x2 2x 3 2 2x 2 x 2 2x 3 2 2x x x2 3 x 2 2x 3 2 x x 2 2x 3 2 2x x x 2 3 x 2 3x x 2 2x3 x x 2 2x3 x xx2 2x3 x 2 2x3 x x 2 3x2xx 2 2x3 x 2 2x3 x x x 2 2x 3 x2 3x 2xx 2 2x 3 x x 2 2x 3 x 3 3x 2 9x 3 x x 2 2x 3 Wht we hve just discovered is tht the function is n ntiderivtive of the function Fx x 2 2x 3 x 2 3 fx x 3 3x 2 9x 3 x x 2 2x 3 Now suppose you sk your friend to do the prolem x 3 3x 2 9x 3 x x 2 2x 3 dx I et they cn t do it But you cn! 3 9
10 3 x 3 3x 2 9x 3 x x 2 2x 3 dx F3 F Assignment: Mke up n integrtion prolem tht you cn do ut tht you think I cn t do Hnd in your prolem (without its solution) on Mondy, Novemer 29 Wht you hnd in on Novemer 29 should hve this formt: Evlute the integrl: 3 x 3 3x 2 9x 3 x x 2 2x 3 dx Then hnd in the solution to your prolem on Decemer 3 (the dy of the next exm) If I cn t do the prolem tht you hnd in (on Novemer 29) nd you show me how to do it (in wht you hnd in on Decemer 3), then I will replce your two lowest prolem scores on the next exm with 0s If I cn do the prolem tht you hnd in on Novemer 29 nd you show me how to do it (in wht you hnd in on Decemer 3), then I will replce your lowest prolem score on the Decemer 3 exm with 0 Hve fun nd e cretive! Note: This is gret opportunity to get good grde on the Decemer 3 exm, ut I expect you to put some effort into it so tht you re getting credit for ctully hving lerned something This ssignment is not for the purpose of getting giving wy free credit in exchnge for only little it of effort from you I expect you to hnd in something tht is very netly written with correct nottion, etc If you hnd in something tht looks like you did not spend much time on it, then I will proly frown upon it 0
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