INTEGRATION IN THEORY

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Stuart McCoy
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1 CHATER 5 INTEGRATION IN THEORY 5.1 AREA AROXIMATION SUMMATION NOTATION Fibocci Sequece First, exmple of fmous sequece of umbers. This is commoly ttributed to the mthemtici Fibocci of is, lthough it is believed tht he did ot ivet it. The sequece begis with 0 d 1, the the ext umber is foud by ddig the previous two. The first severl terms: 0,1,1,,3,5,8,13,1,34,55,89,144,33,377,610,987 if you let A() be the th elemet i the sequece, the A() = A( 1)+ A( ) [i.e., the sum of the previous two elemets. This is exmple of recursively defied sequece. It requires tht you kow A(0) = 0 d A(1) = 1. This sequece is very iterestig, d ivolves the golde-rtio, which ufortutely goes beyod the scope of this course, except to sy tht the sequece hs closed-form defiitio i dditio to the recursive defiitio. The golde-rtio: ϕ = Believe it or ot, it c be show tht A() = ϕ (1 ϕ), test it o your clcultors! SIGMA Σ Whe you must dd my terms, it is ofte coveiet/ecessry to use summtio ottio. We use the Greek letter sigm (Σ) to idicte tht we re tkig summtio. I the previous exmple, we could bbrevite ddig y umber of terms i the Fibocci sequece, lets sy the first 16 terms: = A(0) + A(1) + A() + A(3) + A(4) + A(5) + A(6) + A(7) + A(8) + A(9) + A(10) + A(11) + A(1) + A(13) + A(14) + A(15) = 15 =0 Sice A(0) = 0, why ot just cll it 15 =1 A(). We will study summtios i itricte detil i this course. A() Figure: Fibocci Spirl SUMMATION FORMULAS The fct tht C = C is cler, pretty much from the defiitio of multiplictio, for exmple ++++ = 5. Adrew Dyeso dyesotechig@gmil.com 11

To see tht i= ( + 1), visulize this s dot-coutig: You wt to cout the umber of red-dots. So, you mke cogruet trigle of gree dots iverted, d plced o top of the red-trigle.

2 To see tht i= ( + 1), visulize this s dot-coutig: You wt to cout the umber of red-dots. So, you mke cogruet trigle of gree dots iverted, d plced o top of the red-trigle. The, the result is rectgle, the bse of which hs +1 dots, d the height hs dots. The, there re totl of ( + 1) dots i the rectgle, exctly hlf of which re red. Source: Next, to see tht i = ( + 1)( + 1), see the digrm below: 6 Oe of the pyrmids pictured i the top-left hs the desired volume. Thus, whe you put the three together, slicig hlf of the top piece, d reflectig it, it forms perfect rectgulr prism. The, the followig formul is chieved: X 3 i = ( + 1)( + 1/) Simplifictio yields the desired formul. [4] µ The Evidece tht i3 = i c be see i the followig digrm[4]: Adrew Dyeso dyesotechig@gmil.com 1

The substitute the previous formul X i= Ã X 3 i = 5.1.

3 The substitute the previous formul X i= Ã X 3 i = Adrew Dyeso ( + 1), to chieve the desired result:! i ( + 1) = µ = ( + 1). 4 R IEMANN S UMMATIONS dyesotechig@gmil.com 13

Wht you hve show essetilly is tht the derivtive of the re uder curve fuctio is equl to the fuctio of the curve itself. I other words, if you kow the ti-derivtive, you c fid the re.

4 Figure 5..1: Mrio digitl imge t 800x mgificictio, with zoom-smoothig off 5. THE FUNDAMENTAL THEOREM OF CALCULUS Now tht you hve completed these ctivities, you re redy to derive oe of the most powerful results i Clculus! Wht you hve show essetilly is tht the derivtive of the re uder curve fuctio is equl to the fuctio of the curve itself. I other words, if you kow the ti-derivtive, you c fid the re. Tody I will discuss how these theories fit together to form the Fudmetl Theorem of Clculus. The first clim is the Riem Sum, tht: Recll tht x = b, d x i = + i x. re(f (x), x =, x = b) f (x i ) x The, by the ixel Theory; re(f (x), x =, x = b) = lim b f (x i ) x =: f (x)dx Here the Σ chges to, to idicte refiemet from discrete to cotiuous itervls. For the sme reso, the idividul poits x i chges to cotiuous poits x o the itervl [,b]. Similrly, the ottio for x chges to dx, to idicte tht i the limit x 0. Therefore the dx ecompsses the ide of directio (or vrible) of itegrtio, s well s limit s those rectgles pproch lies. This lso estblishes itimte reltioship betwee Series d Itegrtio, d these two topics will be discussed i detil throughout Clculus II. 1 Next, we let the ed-poit b be vrible, d t this stge we udergo chge i vrible to tht of t, d simply itegrte up to the ed-poit x. Thus the re c be represeted s fuctio of x, which ecompsses the theory of Riem: F (x) := re(f (t), t =, t = x) = f (t)dt f (t i ) t 1 This reltioship is so etwied so tht Series is sometimes referred to s Itegrtio i the literture. Adrew Dyeso dyesotechig@gmil.com 14

5 The mgic hppes whe I set x = t. The, I clim tht F (x + x) is simply oe more rectgle s worth of re further-out i the pproximtio of tht re. The, F (x + x) F (x) ctully pproximtes the re of tht sigle rectge! The re of tht rectgle is f (x) x, the height times the bse. Hese: F (x + x) F (x) f (x) x. Next: F (x + x) F (x) f (x) x Now this is very close to the defiitio of derivtive. Ideed, tkig the limit x 0, this simulteously clcultes the derivtive of the re fuctio, d sice x = t, this lso cuses the Riem pproximtio to become exct, chgig the wvy-equl to equlity: F F (x + x) F (x) (x) = lim = f (x) x 0 x The first prt of the Fudmetl Theorem is stted precisely: Let f be cotiuous fuctio o the itervl [,b], d F (x) = f (t)dt. We kow tht F (x) is cotiuous o [,b] d differetible o (,b). F (x) = d f (t)dt = f (x) dx Wht this mes is tht the re of curve is chgig with rte tht is exctly equl to the vlue of the curve itself. This result hs lmost mysticl qulity bout it. Aother wy of sttig the result is tht the re fuctio is ti-derivtive of the curve fuctio. The secod prt of The Fudmetl Theorem sttes tht if f is cotiuous fuctio o the itervl [, b], d F is ti-derivtive of f o [,b], the we c clculte the re exctly: b f (x)dx = F (b) F () Adrew Dyeso dyesotechig@gmil.com 15

6 The proof of this is bsed o wht we hve lredy show, mely tht F (x) = d dx ti-derivtive of f (x). The we kow tht there is costt C such tht: f (t)dt. Let F (x) be y The, substitute x = ito this fuctio to get: F (x) +C = f (t)dt F () +C = f (t)dt = 0 Sice there is o re withi verticl lie t x = [ xiom of geometry]. But this implies tht C = F (), which yields: F (x) F () = f (t)dt Now, simply substitute x = b ito this fuctio to get the desired result: b f (t)dt = F (b) F () I other words, the re from x = to x = b c be foud by fidig ti-derivtive of the curve fuctio, the evlutig t d b, d subtrctig these two vlues. Adrew Dyeso dyesotechig@gmil.com 16

7 CHATER 6 THE EXONENTIAL The costt e becomes pproched s the umber of compoudigs become lrge. Here, the rte is 100%, d the time itervl t = 1. The, we let grow lrger d see tht the compoudig equtio begis to pproch the idel of cotiuous compoudig by wy of computtio: I Clculus, we will be ble to get-t this umber more exctly. Next, oce we see tht (1 + 1/) e, we c dpt this pproximtio to see tht the compoudig equtio pproximtes cotiuous compoudig for lrge eough. Let r be the desired rte. Replcig with /r is oky, becuse if our rte is less th 100%, the /r becomes lrger sice r < 1, d our clcultio coverges eve fster. O the other hd, if r > 1, the it is true tht /r becomes smller, but I clim tht we oly eed to tke out further i tht cse to get our clcultio to coverge to the desired level of ccurcy. Thus the clim is tht: e r ( ) /r e /r The ext step is to divide 1/(/r ) = r /. Also, tkig the r th power of both sides yields: ( 1 + r ) ( (1 ) ) = + 1 /r r /r Next, tke the t th power of both sides to itroduce time-icremets ito the equtio: ( 1 + r ) t e r t. Filly, multiply both sides of the equtio by the iitil vlue, d we hve successfully derived the cotiuouscompoudig equtio: ( 1 + r ) t e r t 6.1 DEMOIVRE: OLYGONS IN THE COMLEX-LANE Oe of the my my resos tht DeMoivre s theorem is so useful is tht it gives us relly cool wy to progrm polygos. Cosider the exmple of heptgo. Dividig the uit-circle ito seve equl rcs revels ech to be θ = π/7 Adrew Dyeso dyesotechig@gmil.com 17

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)