FunamenalTheorems.nb 1 The Funamenal Theorems of Calculus You have now been inrouce o he wo main branches of calculus: ifferenial calculus (which we inrouce wih he angen line problem) an inegral calculus (which we inrouce wih he area problem). We have also iscovere ha hese wo problems are relae because we have hine a iffereniaion an inegraion as somehow being inverses of each oher. Bu how? We will invesigae ha relaionship in his secion graphically before proceing wih formal efiniions of he Funemenal Theorems. Firs a noe abou he heorems hemselves. If you look a ifferen calculus e books you migh noice somehing curious abou how he Funamenal Theorems are inrouce. Some es presen boh a Firs Funamenal Theorem an a Secon Funamenal Theorem of Calculus (an some es swich he wo!). Ohers will presen one Funamenal Theorem in wo pars (an even when presene as one heorem hey canno agree on which concep is par one an which is par wo). I really oesn' maer o our unersaning of he conceps of he wo heorems (or o he wo pars of one heorem). The main poin is o realize ha he heorems sae ha iffereniaion an (efinie) inegraion are inverse operaions an ha every coninuous funcion is he erivaive of some oher funcion. Le's revisi he angen line problem. Look a he graphs below. The firs one shows he graph of a funcion, f(), an a secan line. fhl Secan line Dy D The slope of he secan line = Dy Å D.
FunamenalTheorems.nb Now look a he secon graph. I shows he same funcion, f(), an a angen line. fhl Tangen line Dy D The slope of he angen line º Dy Å D. We saw in a previous secion how, hrough a limiing process (leing D go o zero) ha he slope of he secan line "becomes" he slope of he angen line. Now le's apply he same analysis o he area uner a curve. Look a he wo graphs below. The firs shows a funcion, f(), an recangle enclosing a porion of he graph of f wih a wih of D an a heigh of Dy. fhl Area of recangle Dy D The area of he recangle = D Dy.
FunamenalTheorems.nb The secon graph shows he same funcion, f(), an he area uner he curve. fhl Area of region uner curve Dy D The area of he region uner he curve º D Dy. Again, we saw in a previous secion ha hrough a limiing process (leing D go o zero) ha he area of he recangle "becomes" he area uner he curve. So we have eermine ha he slope of he angen line is approimaely equal o ÄÄÄÄÄÄÄ Dy an he area uner he curve is approimaely equal o D Dy. I D appears ha since he slope of he angen line is he quoien of Dy an D an he area uner he curve is he prouc of Dy an D, ha iffereniaion an efinie inegraion have an "inverse" relaionship ( a leas in he primiive approimaion sage ). The Funamenal Theorems (co-iscovere by Newon an Leibniz) sae ha he limi processes preserve his inverse relaionship. We are now reay o formally sae he Funamenal Theorems. The Funamenal Theorem of Calculus If a funcion f is coninuous on he close inerval [a, b] an F is an anierivaive of f on he inerval [a, b], hen Ÿ a b f HL = FHbL - FHaL I's ifficul o oversae he imporance of his heorem in he evaluaion of efinie inegrals. If you can fin an anierivaive of a funcion hen you can evaluae a efinie inegral by simply evaluaing he anierivaive a b an a an subracing he resuls!
FunamenalTheorems.nb 4 To help you beer unersan his concep look a he following guielines. Guielines for Using he Funamenal Theorem of Calculus 1. Provie you can fin an anierivaive of f, you now have a meho for evaluaing a efinie inegral wihou having o approimae or use he limi of a sum.. When applying he Funamenal Theorem of Calculus, he following noaion will make he applicaion more convenien. If f is con. on [a, b], an f() = F ' (), hen Ÿ a b f HL = FHL»a b = F(b) - F(a) For insance, o evaluae Ÿ 1, you can wrie Ÿ 1 = 1 = - 1 = 7 - ÅÅÅÅ 1 = 6. I is no necessary o inclue a consan of inegraion, C, in he anierivaive since Ÿ a b f HL = FHL + C» a b = [F(b) + C] - [F(a) + C] =F(b) - F(a). Le's look a some more eamples before moving on o he Secon Funamenal Theorem. Eample 1: Evaluae Ÿ 1 4 è!!! Soluion: Ÿ 1 4 è!!! = Ÿ 1 4 1ê = 4 ÅÅÅ ê ê 1 = H4L ê - H1L ê = 14
FunamenalTheorems.nb 5 Eample : Evaluae Ÿ 0 pê4 sec pê4 pê4 Soluion: Ÿ 0 sec = an» = an ÅÅÅÅ p 4 - an 0 = 1-0 = 1 0 The Secon Funamenal Theorem of Calculus When we efine he efinie inegral of f on he inerval [a, b], we use he consan b as he upper limi of inegraion an as he variable of inegraion. We now look a a slighly ifferen siuaion in which he variable is use as he upper limi of inegraion. To avoi he confusion of using in wo ifferen ways, we emporarily swich o using as he variable of inegraion. Don' le his confuse you. Remember ha he efinie inegral is no a funcion of is variable of inegraion. Le's look a he wo siuaions separaely. The Definie Inegral as a Number Consan a bfhl Consan fisa funcion of The Definie Inegral as a Funcion of F is a funcion of FHL= a fhl Consan fisa funcion of Le's look a an eample of he efinie inegral as a funcion before efining he Secon Funamenal Theorem.
FunamenalTheorems.nb 6 Eample : Evaluae he funcion F() = Ÿ 0 cos a = 0, p/6, p/4, p/, an p/. Soluion: Raher han evaluaing five ifferen efinie inegrals, one for each of he given upper limis, le's fi (as a consan) emporarily an apply he Funamenal Theorem once, o obain Ÿ 0 cos = sin» 0 = sin - sin 0 = sin. Now, using F() = sin, we can obain he following resuls: y FH0L=0 y FH ÅÅÅÅÅ p 6 L= ÅÅÅÅÅ 1 y FH p ÅÅÅÅÅ 4 L= è!!! ÅÅÅÅÅ
FunamenalTheorems.nb 7 y FH p ÅÅÅÅÅ L= è!!! ÅÅÅÅÅ y FH p ÅÅÅÅÅ L=1 F() = Ÿ 0 cos is he area uner he curve f() = cos from 0 o. You can hink of he F() as accumulaing he area uner he curve f() = cos from = 0 o =. For = 0, he area is 0 an F(0) = 0. For = p/, F(p/) = 1 gives he accumulae area uner he cosine curve on he enire inerval [0, p/]. This inerpreaion of an inegral as an accumulaion funcion is ofen use in applicaions of inegraion. Noice in eample above ha he erivaive of F is he original inegran (wih only he variable change). Tha is, ÅÅÅÅÅ @FHLD = ÅÅÅÅÅ @sin D = ÅÅÅÅÅ @ Ÿ 0 cos D = cos We generalize his resul in he following heorem. The Secon Funamenal Theorem of Calculus If f is coninuous on an open inerval I conaining a, hen, for every in he inerval, @ Ÿ a f HL D = f HL
FunamenalTheorems.nb 8 I is also ifficul o oversae he imporance of his heorem. I says ha every coninuous funcion f is he erivaive of some oher funcion, namely Ÿ a f HL. I says ha every coninuous funcion has an anierivaive. An i says ha he processes of inegraion an iffereniaion are inverses of one anoher. You can see why he case can be mae for eiher of he wo Funamenal Theorems o be calle The Funamenal Theorem of Calculus. Le's look a some eamples of applying his heorem. Eample 4: Fin Ÿ p cos an Ÿ 1 1 ÅÅÅÅÅ. 1 + Soluion: Ÿ p cos = cos an Ÿ 1 1 ÅÅÅÅÅ = 1 + 1 1 + Eample 5: Fin y/ if y = Ÿ 1 sin. Soluion: This one is a lile more complicae han hose in eample 4 because he upper limi of inegraion is no bu. This makes y a composie of y = Ÿ 1 u sin an u = So we mus apply he Chain Rule when fining y/. y = y u ÿ u =( u u Ÿ 1 sin ) u = sin u u = sinh L ÿ = sin Wha happens if he lower limi is variable an he upper limi is a consan? We jus nee o apply a propery of efinie inegrals o make an ajusmen o he inegral. Look a he eample below.
FunamenalTheorems.nb 9 Eample 6: Fin y/ given y = Ÿ e Soluion: Ÿ e = H- Ÿ e L = - Ÿ e = - e Wha if boh he lower limi an upper limi are variable? We jus use some of he oher properies of efinie inegrals as in he eample below. 1 Eample 7: Fin y/ given y = Ÿ ÅÅÅÅÅ + e Soluion: Ÿ 1 0 1 ÅÅÅÅÅ = + e J Ÿ ÅÅÅÅÅ + + e Ÿ 1 0 ÅÅÅÅÅ N + e = J- 1 Ÿ 0 ÅÅÅÅÅ + 1 + e Ÿ 0 ÅÅÅÅÅ N + e = - 1 ÅÅ + e H L + 1 Å + e H L = - ÅÅ ÅÅ + + e + e So how is he Secon Funamenal Theorem useful in our suy of calculus? Remember ha no all funcions have elemenary anierivaives. I's easy o be foole ino hinking so since we spen so much ime solving ifferenial equaions using separaion of variables. The number of funcions ha acually have an elemenary anierivaive is acually quie small. The Secon Funamenal Theorem provies a way of ealing wih hose ha on'. Look a he ne eample. Eample 8: Fin a funcion y = f() wih erivaive y = an ha saisfies he coniion f() = 5.
FunamenalTheorems.nb 10 Soluion 1: We can o his problem wihou using he Secon Funamenal Theorem which is he meho we presen firs. We jus have o use a lile subsiuion. y = Ÿ an = Ÿ ÅÅÅÅ sin cos We now le u = cos an u = -sin, his resuls in -Ÿ 1 ÅÅÅÅ u u= - ln u + C or y = - ln cos + C Now using he coniion f() = 5 we can solve for he consan C. So we have 5 = - ln cos + C ï C = 5 + ln cos y = - ln cos + ln cos + 5 or y = ln» cos ÅÅÅÅ cos» + 5 This was a lo of work. Le's use he Secon Funemenal Theorem. Soluion : Le's le y = Ÿ an. Since y() = 0 (i.e. Ÿ an = 0), we nee o a 5 o his funcion o consruc one wih erivaive an whose value a = is 5: f() = Ÿ an + 5 Alhough his funcions saisfies he wo original coniions you migh well ask wheher i is very useful. Since we saw in a previous eample ha a funcion in his form can be hough of as an accumulaion funcion, wih he ai of a graphing calculaor or compuer, i can be evaluae quie simply for any value of. I was cerainly easier o fin he funcion using his meho. An i works jus as well on hose funcions ha o no have an elemenary anierivaive as his one i. I chose his eample so we coul
FunamenalTheorems.nb 11 compare he wo mehos. By now you mus be iching for some applicaions of efinie inegrals now ha you are arme wih a powerful way o evaluae hem. Tha is wha he ne secion is all abou. Workshee 1a: Evaluaing a Definie Inegral Workshee 1b: Funcions Define as Definie Inegrals Workshee 1c: The Secon Funamenal Theorem