12.2 The Definite Integrals (5.2)

Transcription

1 Course: Aelerted Egieerig Clulus I Istrutor: Mihel Medvisky. The Defiite Itegrls 5. Def: Let fx e defied o itervl [,]. Divide [,] ito suitervls of equl width Δx, so x, x + Δx, x + jδx, x. Let x j j e ritrry smple poits suh tht x j x j, x j. The the defiite itegrl of f from to is f xdx itegrle o [,]. Notes: f x j Δx provided tht the limit exists. If it douse exist, we sy tht f is is itegrl sig, fx is itegrd d, re lower d upper limits of the itegrl respetively. Evlutig\lultig the itegrl is lled itegrtio. The itegrl is ot depeded o x, i.e. f xdx f tdt f rdr If fx> i [,], the itegrl represet the re tht lies uder fx. For fx< i [,], the itegrl represet A of -fx. If f hges sigs, the it represet the differee etwee res of egtive d positive regios. The Reim sum e equivletly defied usig itervls of uequl width. Ex. Evlute x dx. Solutio: Sketh the grph of x- d figure out tht the futio is rossig x-xis, therefore the itegrl is the differee of res of trigles. Thus x dx.5. Thm: If f is otiuous o [,], or if it hs fiite umer of jump disotiuities, the f is itegrle. Thm: If f is itegrle o [,] the x, x + Δx, x j + jδx, x. f xdx f x j Δx where Δx d

2 Course: Aelerted Egieerig Clulus I Istrutor: Mihel Medvisky.. Evlutig Itegrls Ex. Ex 4. Ex 5. dx dx x dx Δx + Δx x j Δx j Δx x dx Δx Δx + jδxδx Δx + j Δx Δx + Δx x j Δx Δx + j Δx + j Δx + lim jδx Δx + + lim Δx + lim Δx + Δx jδx + jδx Δx + + Δx + + lim Ex. x dx x j Δx + jδx Δx + jδx + jδx + jδx Δx Δx + j Δx + j Δx + j Δx

3 Course: Aelerted Egieerig Clulus I Istrutor: Mihel Medvisky.. Midpoit Rule It is sometime usle to estimte itegrl umerilly. Isted of evlutig limits oe hoose fiite smll d x j x j + x j, the use the defiitio. Ex 7. The ext solutio, s show i previous exmple is x dx " +/ $ + " / + $ + " + / $ + " / + $ # # # # " $ + " $ + " 5 $ + " 7 $ " $ # 4 # 4 # 4 # 4 # 4 8 Ex 8. Approximte si x dx usig Midpoit Rule, ompre to ed poit rule k x j x j. si x dx si si x dx si si x dx si 4 + si 4 +. si x dx si + si Properties of Defiite Itegrl. f xdx f x j Δx. f xdx f x j. dx 4. f x j Δx f x j lim f x j Δx dx { f x ± g x }dx { f x j ± g x j }Δx 5. f xdx lim f x j f xdx f x j Δx ± lim g x j Δx f xdx ± g xdx f x j Δx lim f x j Δx f xdx

4 Course: Aelerted Egieerig Clulus I Istrutor: Mihel Medvisky. f xdx f xdx + f xdx Ex 9. dx + dx x dx + x dx + # $ 7. If f x, x [, ] f x 8. dx + dx f x g x, x [, ] f x g x # $ x dx f x g xdx f xdx g xdx f xdx g xdx 9. m f x M mdx f xdx M dx m f x dx M Ex. Estimte x x +dx.9 : We would like to oud the futio, i.e. we first fid glol mi d mx vlue of the itegrd. Let f x x x +, the first derivtive test give us f x x x x,, we ext ompre vlue of f t these d ed poits. f ; f ; f.85 f x.85 f x dx. Evlutig Defiite Itegrls 5.! $ # " Thus Evlutio theorem: Let f e otiuous o [,] d let F e ritrry tiderivtive of f, i.e. F f, the f xdx F F I order to see why the theorem ove works we osider dividig [,] ito suitervls with the followig ed poits x, x,..., x d Δx x j+ x j. We F x j ext use Me Vlue Theorem to see tht F x j x j x j f x j where x j #$ x j, x j,

5 Course: Aelerted Egieerig Clulus I Istrutor: Mihel Medvisky thus F x j F x j f x j Δx. Therefore: F F F x F x F x F x + F x F x + F x F x f x Δx f x Δx f x j Δx Note: F F F F f x j Δx f xdx. Nottio: f xdx F x F F Ex. Ex. dx x x dx x + Ex. xdx x Ex 4. Ex 5. + m m si x dx os x os os os + e t ost e t sit + e t si t + e t ost l dt e t ost sit + si t + ost dt l l + os t + ost sit + si t dt e t os t ost sit + si t l e t dt e t dt e t l l.. Idefiite Itegrl Def: The ommo ottio of tiderivtive is f xdx f xdx F x mes F x f x Note tht the differee etwee idefiite itegrl f xdx d defiite itegrl f xdx is tht the former is futio wheres the lter is umer. The oetio is give y Evlutio theorem: f xdx f xdx

6 Course: Aelerted Egieerig Clulus I Istrutor: Mihel Medvisky Tle of idefiite itegrls:. f xdx f xdx. f x + g xdx f xdx + g xdx. x dx + x+ +C 4. x dx l x +C 5. e x dx e x +C. x dx x l +C 7. os x dx si x +C 8. si x dx os x +C os x dx t x +C si x dx ot x +C x + dx t x +C dx si x +C x Ex. x + 7 dx x dx + 7 dx x + 7x +C Ex. Ex. Ex 4. x +5 dx 9x + x + 5 dx x dx +5 x dx + 5 dx x +5x + 5x +C dx + osx dx os x t x +C x x x os si si xdx dx + C 4 osx os 5x x xdx x + x dx + C Ex 5. si os si si 5 Ex. t t + t + t t +C os t