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1 Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht you cn lern here: The prcticl version of the FTC. No kidding! The theoreticl version of the FTC gives us nice piece of informtion out the re function nd the definite integrl tht defines it, ut still does not tell us how to compute either. Or, to sy etter, does not tell us eplicitly. To do tht, we only need to turn its sttement round it. Technicl fct Fundmentl Theorem of Clculus: Prcticl version of the FTC Given function y f tht is continuous on the intervl,, nd given ny ntiderivtive F of f, then: f ( ) d F( ) F( ) Proof A f. But this mens tht A must e n ntiderivtive of The theoreticl version of the FTC tells us tht the derivtive of ut which of the infinitely mny ntiderivtives is it? To find out, let s pick ny one ntiderivtive, sy is f, F, nd notice tht, f to e continuous, A must differ from F only y n dditive constnt: A F c since we re ssuming But we lso know t lest one vlue of A ( ) A f t dt, since we know tht: Therefore F( ) c, c F( ) nd A( ) F( ) F( ). From this follows tht: f () t dt A F F Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7: The FTC prcticl version Pge
2 Needless to sy, ut I ll sy it nywy, since we hve choice of which ntiderivtive to use in the ppliction of the FTC, we choose the one whose constnt of integrtion is c, tht is, we don t other dding the constnt. I dmit tht this is cool nd I enjoyed the FTC in high school, when we were first told out it. But why is it clled Fundmentl? Very legitimte question, since in mthemtics we don t cll ny sttement fundmentl, or centrl, or importnt, ut only those tht deserve the title. So, here is why. Knot on your finger The Fundmentl Theorem of Clculus (FTC) is clled Fundmentl ecuse: It llows us to solve the re prolem in its generlity, s we shll see soon. It solves the re prolem in very computtionlly efficient wy. It shows tht two very importnt prolems in mthemtics, finding slopes nd finding res, tht seemed unrelted, re in fct strictly linked. Since the sme ide of definite integrl used to solve the re prolem cn e pplied to mny other geometricl nd physicl prolems, the FTC llows us to solve mny more importnt prolems in the physicl sciences. Such solutions hve llowed, in recent time, the development of the high level of technology tht we enjoy tody. I think tht this is eciting: mny more doors re now open for us to eplore. Are we going to see those uses soon? Very soon, ut first let us fi detil in the nottion nd look some more t how elegntly the FTC solves the re prolem. If Knot on your finger Definite integrl nottion F is the ntiderivtive of y f tht we use to pply the prcticl version of the FTC, we write, s n intermedite step imed t clerly identifying such ntiderivtive: f ( ) d F F( ) F( ) Other, similr nottions cn lso e found in the literture, such s: Emple: f ( ) d F F( ) F( ) y y sin,,, These three curves ound the region in the first qudrnt under the given function nd to the left of. Its re is therefore defined y sin d If we use the definition of integrl, we would need to compute: n lim sin i n i Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7: The FTC prcticl version Pge
3 Even if we ssume ll rectngulr slices to hve the sme length, so tht ll re equl, this is n impossile limit to figure out. But, thnks to the i FTC, its computtion oils down to computing n ntiderivtive of y sin. By using integrtion y prts you cn check tht this is:: The rest is now esy y using the FTC: sin d sin c sin d sin.57 The FTC is simple enough tht further emples should e unnecessry. The lerning questions will provide mple illustrtions nd eperience for you. Moreover, we shll see mny pplictions of this theorem nd, with them, mny emples of how to use it. If you do need further worked out emples, there re mny we sites tht provide them. Ain t it gret when you re studying something difficult, ut well known nd used? And, speking of pplictions nd uses, here is first, very pplied nd very generl consequence of the FTC. If quntity Technicl fct The net chnge theorem Qt chnge continuously during the time period from to, then the net chnge in this quntity during this period is given y: ' Q Q Q t dt Proof This is just the sttement of the prcticl version of the FTC, ut with n eqully prcticl interprettion! This fct is prticulrly useful when the function descriing the quntity is not known, ut its rte of chnge is, s we sw when studying ODE s. Emple: v( t) t t If n oject is moving on the yis so tht its velocity is given y this function, its net chnge in the position for the first five seconds is given y: 5 5 t 5 y(5) y() t t dt t Notice, however, tht this is not the totl distnce trvelled y the oject during this time, since the oject my hve moved ck nd forth within tht time period. In fct, since: t t t t we cn see tht the oject trvels down from t to t, ut trvels up fter t. Therefore, s we did for computing totl re, the totl distnce it trvels is: 5 T t t dt t t dt 5 t t 58 t t So, this is the first ppliction of definite integrl outside of res, nd s it efits method tht ws invented to descrie the motion of plnets, it reltes to moving ojects. Mny more uses to come, just sty tuned nd ecited! Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7: The FTC prcticl version Pge
4 Summry A definite integrl whose integrnd is continuous my e computed s the difference etween the vlues of one of its ntiderivtives t the two endpoints. This fct provides prcticl solution to wide vriety of re prolems nd will provide the generl solution to this prolem, s well s to mny other pplied prolems. In prticulr, the prcticl version of the FTC llows us to esily compute the net chnge of quntity who rte of chnge is known. Common errors to void The FTC looks nd is esy to understnd nd implement. But don t forget tht ehind it there is lrge mount of lger nd indefinite integrtion tht must e done correctly in order for the theorem to work correctly. Lerning questions for Section I 47 Review questions:. Eplin wht the prcticl version of the FTC sttes.. Descrie wht the net chnge theorem llows us to do. Memory questions:. Wht does the prcticl version of the FTC stte?. Which one condition on the integrnd is required y the prcticl version of the FTC? Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7: The FTC prcticl version Pge 4
5 Computtion questions: In ech of questions 6, use the prcticl version of the FTC to evlute the given definite integrl.. 4 d. 4 sin d 5. / 5 d. sin e d 4. ln d 6. tn d In ech of questions 7 compute the re of the region ounded y the given curves. 7. y sin, y,,. 8. y e, y,,. sin if 9. y if. y e y,,, nd the is. t. Compute the derivtive of the function h() t d in the following two wys nd verify tht they provide the sme nswer: ) evluting the integrl nd then differentiting it. ) Applying the theoreticl version of the FTC.. Compute ech of the following seemingly similr epressions involving integrls. ) sin e d ) sin d d e d c) sin e d d) sin Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7: The FTC prcticl version Pge 5 d d d d e d t e) sin t e dt
6 . Determine the net chnge of the quntity Q, whose derivtive is Q' e, etween nd. In your nswer clerly ehiit ll mjor steps nd identify the one where the FTC is used. 4. Determine the net chnge of the quntity Q, whose derivtive is Q' sin, etween nd ll mjor steps nd identify the one where the FTC is used.. In your nswer clerly ehiit Theory questions:. Cn we use the prcticl version of the FTC to compute ln d. Wht is the reltion etween indefinite nd definite integrls? 4. Stte two resons why the FTC is clled fundmentl.. Use the FTC to compute the re of the region ounded y the curves y, y,, work?. How do you eplin the conclusions of your Proof questions:. Use the FTC to determine the vlue of d. How do you eplin the nswer you get, ssuming you get the correct one? Appliction questions:. If m' f ( v) represents the rte t which the mss of n oject chnges s its speed v chnges (ccording to the theory of reltivity), wht does f () v dv represent? In the cgs system, wht re units of f() v? Wht questions do you hve for your instructor? Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7: The FTC prcticl version Pge 6
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