Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]

Transcription

1 Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1

2 The Bsc Ide We the crete rectgles f (x) x-xs 4 The Bsc Ide We exme ech sutervl oe t tme, fdg the re of rectgle f(x) A = f (x) x x = umer of su-tervls We the sum the res of ech rectgle 5 Lower Estmte We estmte the re usg scred rectgles = 4 6

3 Upper Estmte We estmte the re usg crcumscred rectgles = 4 7 Mdpot Estmte = 4 8 Remrks L Lower Estmte U Upper Estmte mare M We c crese the ccurcy of our estmte y: Avergg m d M By usg more rectgles Rectgles do ot hve to e the sme wdth 9

4 Exmple f ( x) x o 1, We wll use oe rectgle to estmte the re 10 Lower Estmte L 5(1) 5 11 Upper Estmte U 14(1)

5 Exmple f ( x) x o 1, We wll use two rectgles to estmte the re 1 Lower Estmte L Upper Estmte U

6 Remrks Compre our results L U (L+U)/ Add oe more le for comprso As creses we pproch the ctul re = 9 16 Useful Equtos o the Prtto Gve f ( x) o, d (umer of su-tervls) wdth of su-tervl frst x secod x th x x x0 1 x x x x 1 x lst x 17 Remrks As creses (more rectgles) our ccurcy mproves Iscred rectgles result low estmte L Crcumscred rectgles result hgh estmte U L Actul Are U Rectgles do ot hve to hve equl wdths Could use verge vlue sted of the left or rght fucto vlue C use trpezods to mprove estmte 18 6

7 Georg Fredrch Berhrd Rem Rem's des cocerg geometry of spce hd profoud effect o the developmet of moder theoretcl physcs. He clrfed the oto of tegrl y defg wht we ow cll the Rem tegrl. 19 Rem Sum We crete rtrry rdomly selected prtto of the tervl to sutervls x0 x1... x 1 x We fd the re of f( x) o, y summg rectgles x0 x1 x x 1 x 0 Rem Sum x0 x1 x x 1 x x x x x x x The sutervls re,,,,..., wth wdths x, x,..., x where x 1 x x 1 The lrgest wdth s clled the orm d s deoted x 1 7

8 Rem Sum we the choose smplg pots from ech su-tervl c x, x 1 c s x, x c s x, x 1 c s x, x c s x, x 1 The Rem Sum s 1 f c x The Rem Sum s Remrks f cx 1 The Are s lm f cx x 0 1 x x Our ccurcy mproves s x the orm pproches zero If we use tervls of equl wdth x - x x 0 But does ot mply x 0 Remrks Oe lrge rectgle My smll rectgles 4 8

9 Steps for Rem Sums ove, Fd re of rego ouded y f x x o the left, x o the rght d the x-xs elow For ese of computto we geerlly use prttos of equl sze Beg y computg the rectgle wdth Idetfy the frst x usg the left-edpot of the tervl x x0 5 Steps for Rem Sums Compute x x x0 x Clculte the heght of oe rectgle f x Fd the re of oe rectgle f x x Add the rectgles 1 f x x Clcultos re eser f we smplfy t ech step 6 f ( x) x o 1, 4 Exmple rectgle wdth Frst x 1 1 x 4 4 x0 1 x x x x Heght of oe f x f 1 4 rectgle

10 Exmple Are of oe rectgle f ( x) x o 1, 4 1 f x x Sum of the rectgles 1 f x x Dstrute Exmple f ( x) x o 1, 4 Sum usg the summto formuls Smplfy Exmple f ( x) x o 1, 4 We used rght sum We could hve used left sum Ad the ctul re s A

11 Exmple f ( x) x o 1, 4 y=4 y= y= 1 Are of rectgle & trgle s ()(1) + (1/)(1)(1) = ½ =.5 1 Exmple The frst x Wdth of the rectgle x Heght of oe rectgle f ( x) x o,6 10 x0 6 x x 10 f x Exmple Are of rectgle Rght Sum of rectgles Dstrute f ( x) x o, f x x f x x

12 Exmple Rght Sum f ( x) x o, f x x Left Sum f x x Actul re s Steps for Rem Itegrl ove, Fd re of rego ouded y f x x o the left, x o the rght d the x-xs elow For ese of computto we geerlly use prttos of equl sze Beg y computg the rectgle wdth Idetfy the frst x usg the left-edpot of the tervl x x0 5 Steps for Rem Sums Compute x x x0 x Clculte the heght of oe rectgle f x Fd the re of oe rectgle f x x Add the rectgles 1 f x x Tke the lmt lm f x x 1 6 1

13 Exmple f ( x ) x o 1, rectgle wdth Frst x 1 1 x x0 1 x x x x 0 1 Heght of oe f x f 1 rectgle 1 7 Exmple Are of oe rectgle f ( x) x o 1, 1 f x x Sum of the rectgles f x x 1 1 Dstrute Exmple Sum Tke lmt f ( x) x o 1, lm lm

14 Exmple f ( x) x o 1, 7 Rght Sum lm 1 Left Sum 1 7 lm 0 Whe we tke the lmt we get the result regrdless of whch c we use ech tervl 40 The Defte Itegrl Let f ( x) e fucto defed o closed tervl, We sy tht I s the defte tegrl of f over, d I s the lmt of the Rem Sum lm f c x f the followg codto s stsfed: k k k 1 Gve y umer >0 there s correspodg P x0 x1 x P x x x umer 0 such tht for every prto,,, of, wth d y choce of,, we hve f ckx k I 41 k 1 k k 1 k The Defte Itegrl f x Whe we fd the re of ( ) o, k 1 k k we wrte I lm f c x f x dx We refer to s the lower lmt of tegrto s the upper lmt of tegrto f x dx s the Defte Itegrl Red s the tegrl of f from to 4 14

15 Remrks Whe we foud the tdervtve we foud the defte tegrl d wrote resultg fmly of fuctos where the defte tegrl umer represetg the re f x dx f x dx results 4 Exmple Usg m mxdx 4 4xdx xdx 44 Propertes of Defte Itegrls c f x dx f x dx f x dx c c + = 45 15

16 Propertes of Defte Itegrls cf x dx c f x dx Ths mes we c dstrute out costt 46 Propertes of Defte Itegrls f x g x dx f x dx g x dx Ths mes we c rekup sum or dfferece to smller tegrls 47 Propertes of Defte Itegrls f x dx f x dx Ths mes f we swtch the lmts we chge the sg of the tegrl 48 16

17 Propertes of Defte Itegrls 0 f x dx Cosder rectgle wth legth (x) d wdth (y). The re s the A = x y Clerly f the wdth y = 0 the the re A = 0 49 Exmples 7 1 Gve f x dx 5 d f x dx Fd f x dx Fd f x dx 1 Fd f x dx 0 = 5 1 = f x dx 50 17