Approximations of Definite Integrals

Transcription

1 Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl f(x)dx, stdrd procedure hs bee to look for tiderivtive F (x) for the itegrd f(x) the use the fct tht f(x)dx = F (b) F () We hve bee fortute i tht every itegrd f(x) we hve fced, to this poit, hs "fmilir" tiderivtive. The bd ews is tht there re my useful fuctios with o such tiderivtives. A exmple is f(x) = e x, which is very commo fuctio i probbility theory. Also, there re fuctios tht re give s tbles, (from experimetl dt), with o formul describig the reltioship betwee the vribles. I cses like these, we hve to settle for "good" pproximte vlues of their defiite itegrls. To clculte them, we visulize f(x)dx s the re of the regio eclosed by the x-xis d the grph of f o the itervl [, b], the look for pproximte vlues of the re. The stdrd procedure is to divide it ito smller strips by dividig the itervl [, b] ito smller subitervls [x, x ], [x, x ],..., [x, x ] of width h = (b ) ech. (Thus x =, x = + h, x = + h,..., x = + h = b.) The regio Regio prtitioed ito strips Ech strip is the pproximted with geometric figure whose re is fmilir. The stdrd oes re rectgles, trpeziums; i geerl, figures eclosed by polyomils. We strt with rectgles. Usig Rectgles of Equl Width We pproximte the strip o itervl [, ] with rectgle tht hs bse [, ] d pproprite height H i. There re vrious choices for H i :. We choose the height H i to be the vlue f( ) of f t the left-ed poit of the itervl [, ]. The f(x)ds pproximted by the sum of the res hf(x ), hf(x ),..., hf(x ). Sice h = (b ), (b ) [ ] f(x ) + f(x ) + + f(x )

2 This is clled the left-edpoit rule, becuse it results from pproximtig the strips with leftedpoit rectgles. Left-edpoit rectgles. We choose H i to be the vlue f( ) of f t the right-ed poit of the itervl [, ]. The is pproximted by the sum of the res hf(x ), hf(x ),..., hf(x ). Usig h = (b ), (b ) [ ] f(x ) + f(x ) + + f(x ) f(x)dx This is clled the right-edpoit rule, becuse it results from pproximtig the strips with rightedpoit rectgles. Right-edpoit rectgles. We choose H i to be the vlue of f t the midpoit θ i = ( + ) of the itervl [, ]. The f(x)ds pproximted by the sum of the res hf(θ ), hf(θ ),..., hf(θ ). Sice h = (b ) (b ) [f(θ ) + f(θ ) + + f(θ )] This is clled the midpoit rule, becuse it results from pproximtig the strips with midpoit

3 rectgles. Exmple To pproximte Midpoit rectgles e x dx usig 5 rectgles of the types described bove. Here the itegrd is f(x) = e x. The itervl [, ] is divided ito the 5 subitervls [ [, 5], [ 5, 5], 5, ] [ 5,..., 5, ]. Left-edpoit Right-edpoit Midpoit If we pproximte the strips with left-edpoit rectgles the e x dx 5 f() + 5 f ( ) f ( ) f ( ) 5 Pull out clcultor d evlute [ ( 5 e + e /5 + e /5 + + e /5)]. The result should be.66, (to deciml plces). If we pproximte the strips with right-edpoit rectgles we get to deciml plces. e x dx 5 f ( ) f ( ) f ( ) f ( 5 5) =.776 To pproximte the strips with midpoit rectgles it is ecessry to determie the mid-poits of ech itervl. It is for the itervl [, 5], for the itervl [ 5, ] 5,..., 9 for the lst itervl. Therefore, to deciml plces, e x dx 5 f ( ) + 5 f ( ) + 5 f ( ) f ( 9 ) =.77

4 Usig Trpeziums of Equl Width This time we pproximte the strip o itervl [, ] with the trpezium joiig the four poits (, ), (, f( )), (, f( )) d (, ). It hs re (f( ) + f( )) h. Therefore h [f(x ) + f(x ) + f(x ) + f(x ) + + f(x ) + f(x )] () With the exceptio of f(x ) d f(x ), every other term iside the squre brckets ppers twice, therefore (b ) This is clled the trpezoidl rule. [f(x ) + f(x ) + f(x ) + + f(x ) + f(x )] Approximtig trpeziums Strip Trpezium Exmple A pproximtio of to deciml plces. e x dx Usig Qudrtic Curves e x dx usig 5 trpeziums is ( e + e /5 + e /5 + e /5 + + e /5 + e 5/5) =.79 5 I this cse we must prtitio the itervl [, b] ito eve umber of subitervls [x, x ], [x, x ],..., [x, x ]. We the pproximte the two strips o itervl [, + ] with the regio eclosed by the qudrtic curve q i (x) tht psses through the three poits (, f( )), (+, f(+ )) d (+, f(+ )), (see the figures below). Two strips By Exercise?? o pge??, the re uder the qudrtic curve is xi+ q i (x)dx = (+ ) 6 = (+ ) 6 Approximtig qudrtic [q( ) + q(+ ) + q(+ )] [f( ) + f(+ ) + f(+ )].

5 (b ) Note tht (+ ) =, hece (+ ) = (b ). Therefore 6 Similrly, x x x x x x x q (x)dx = x q (x)dx = (b ) [f(x ) + f(x ) + f(x )]. (b ) [f(x ) + f(x ) + f(x )]. I geerl, xi+ xi+ q i (x)dx = (b ) [f( ) + f(+ ) + f(+ )]. Addig up gives (b ) {f(x ) + f(x ) + f(x ) + f(x ) + f(x ) + f(x )+ + f(x ) + f(x ) + f(x )} This simplifies to the followig Simpso s rule. (b ) {f(x ) + [f(x ) + f(x ) + + f(x )] + + [f(x ) + f(x ) + + f(x )] + f(x )} This my look complicted but it is esy to remember. Add the vlues of f t the poits x, x,..., x with eve idexes the multiply the result by. This gives you the term [f(x ) + f(x ) + + f(x )]. Next dd the vlues of f t the poits x, x,..., x with odd idexes d multiply the result by. This gives the term [f(x ) + f(x ) + + f(x )]. To these two terms, dd the vlue of f t x d x, (i.e. (b ) the vlue of f t the ed-poits of the itervl). Filly multiply the result by. Exercise Let I ech of the followig questios, roud off your swers to deciml plces.. Use the midpoit rule, with = 8, to estimte. Use the trpezoidl rule, with = 8, to estimte. Use the left-edpoit rule, with = 5, to estimte + x dx.. Use the right-edpoit rule, with = 5, to estimte 5. Use Simpso s rule with = 8 to estimte 6. We kow tht e x dx. e x dx. π/ + x dx = rct rct = π. () Use the trpezoidl rule, with =, to estimte π. si(x )dx. + x dx. (b) Use the midpoit rule, with =, to get other estimte of π. (c) Use the Simpso s rule, with =, to get yet other estimte of π. 5

6 Errors i Estimtes of Defiite itegrls As before, we view f(x)dx s the re of the regio R eclosed by the x-xis d the grph of f o the itervl [, b]. Assume tht f is differetible o [, b]. Prtitio the itervl ito subitervls [x, x ], [x, x ],..., [x, x ] where x =, x = x + h,..., x = x + h = b, of legth h = (b ) ech. The R is prtitioed ito strips with res x x f(x)dx, x x f(x)dx,..., x x f(x)dx Sy we pproximte f(x)dx with left-edpoit rectgles. This mes tht the re ith strip, is pproximted with the vlue f( )h. Note tht we my write this s f( )h = xi+ f(x)dx of the xi+ f( )dx. We wish to estimte the error i the pproximtio. The bsolute vlue of the differece betwee the exct vlue d the pproximtio is xi+ [f(x) f( )]dx. For ech x betwee d +, the Me Vlue Theorem ssures us tht there is umber θ x betwee x d such tht f(x) f( ) = (x ) f (θ x ) Let M be the lrgest vlue of f (x) o the itervl [, b]. The (x ) f (θ xi ) M (x ) d so xi+ xi+ [f(x) f( )]dx = (x ) f (θ x )dx Thus the error i pproximtig xi+ M xi+ f(x)dx with f( )h = up, we coclude tht the error i pproximtig does ot exceed Mh + Mh + + Mh We hve used the fct tht h = (b ). Exmple Cosider [ (x ) dx = M (x ) xi+ ] xi+h = Mh. f( )dx does ot exceed Mh. Addig f(x)dx with the res of the left-edpoit rectgles ( ) Mh = = M(b ) = dx. The derivtive of f(x) = + x + x is f x (x) = ( + x ). M(b ). O the itervl [, ], the lrgest vlue of the umertor s d the smllest vlue of the deomitor is. Therefore the lrgest vlue of f (x) o the itervl does ot exceed =. It follows tht the error i pproximtig dx with the res of left-edpoit rectgles does ot exceed + x =.. Sice the totl re of the rectgles is pproximtely.8998, d + x dx = π, we coclude tht π =.8 ±., or π =. ±. 6

7 We stte the ext three error estimtes without proof:. Suppose f (x), the secod derivtive of f(x), is cotiuous, d f (x) M for ll [, b]. The M(b ) the error i pproximtig f(x)dx with midpoit rectgles does ot exceed. Exmple 5 Cosider dx. The secod derivtive of f(x) = + x + x is f (x) = ( + x ) + 8x ( + x ). The vlue of f (x) o [, ] does ot exceed + 8 =, (pply the rgumet i exmple () to ech term). Therefore the error i pproximtig dx with the re of midpoit rectgles does + x ot exceed.. The re of the midpoit rectgles is pproximtely.7856, it follows tht π =.7856 ±., or π =. ±.. Suppose f (x), the secod derivtive of f(x), is cotiuous, d f (x) M for ll [, b]. The M(b ) the error i pproximtig f(x)dx with trpeziums does ot exceed.. Suppose f () (x), the fourth derivtive of f, is cotiuous, d f () (x) M for ll [, b]. Let [, b] be divided ito eve subitervls of legth (b ) / ech. The the error i pproximtig M(b )5 f(x)dx usig Simpso s rule does ot exceed 8. Exercise 6. Let f(x) = cos x, x. () Use the left-edpoit rule with = 6 to estimte i your swer. (b) Use the mid-poit rule with = 6 to estimte i your swer. (c) Use the trpezoidl rule with = 6 to estimte i your swer. (d) Use Simpso s rule with = 6 to estimte your swer. cos x dx. Also determie the mximum error cos x dx. Also determie the mximum error cos x dx. Also determie the mximum error cos x dx. Also determie the mximum error i. Determie vlue of so tht the mid-poit rule pproximtes th.. + x dx with error of less 7