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
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
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
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) 14 1 4
Exmple f ( x) x o 1, We wll use two rectgles to estmte the re 1 Lower Estmte 1 5 1 L 5 4 55 8 14 Upper Estmte 5 1 1 U 14 4 91 8 15 5
Remrks Compre our results L U (L+U)/ 1 5 14 9.5 6.875 11.75 9.15 Add oe more le for comprso 4 7.9065 10.156 9.015 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
Georg Fredrch Berhrd Rem 186-1866 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,,,,..., 0 1 1 1 wth wdths x, x,..., x where x 1 x x 1 The lrgest wdth s clled the orm d s deoted x 1 7
Rem Sum we the choose smplg pots from ech su-tervl c x, x 1 c s x, x 1 0 1 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
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 0 1 4 Heght of oe f x f 1 4 rectgle 1 4 7 9
Exmple Are of oe rectgle f ( x) x o 1, 4 1 f x x 4 4 16 4 Sum of the rectgles 1 f x x 4 1 16 4 Dstrute 1 16 4 4 4 1 1 1 8 Exmple f ( x) x o 1, 4 Sum usg the summto formuls 1 4 4 1 4 16 4 1 Smplfy 9 8.65 9 Exmple f ( x) x o 1, 4 We used rght sum We could hve used left sum 4 1 0 9.65 16 4 8 7.75 16 4 8 Ad the ctul re s A.5 0 10
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 10 10 x 10 f x 10 10 Exmple Are of rectgle Rght Sum of rectgles Dstrute f ( x) x o,6 10 9 f x x 10 10 100 f x x 10 1 1 10 9 100 1 9 100 11
Exmple Rght Sum f ( x) x o,6 10 1 9 10 10 1 f x x 100 99 4.95 0 Left Sum 9 9 81 f x x 4.05 100 0 1 0 Actul re s 4.5 4 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
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 1 1 1 1 8 Exmple Sum Tke lmt f ( x) x o 1, 1 1 1 1 1 1 lm lm 1 7 9 1
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
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 9 4 10 4xdx 44 Propertes of Defte Itegrls c f x dx f x dx f x dx c c + = 45 15
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
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 4 7 1 Fd f x dx 5 4 9 Fd f x dx 1 Fd f x dx 0 = 5 1 = f x dx 50 17