7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg lke x e dx to hgh level of ccurcy? I chpter 5 you lered tht f we dvde, to sutervls of legth * we hve ( ) f x dx f x x where, x (Note: wth ths mg scheme we hve x x, the * x s y pot the th sutervl whch we cll x x for 0,,..., ). You lso lered tht the mdpot rule s the pproxmto tht we ot whe we let * x x x. The mdpot rule s pctorlly represeted elow wth 4 : x 0 x x x 3 x 4 If the curve purple represets the grph of f( x ) the the mdpot rule tells us tht the pproxmte vlue of x4 dx s gve y the sum of the res of the four yellow rectgles. x0
Alytclly we c represet the mdpot rule y: dx x f x f x... f x where x x x d x. Clerly the pproxmto yelded y the mdpot rule gets etter s the vlue of creses. A good questo to sk s: Is there cert vlue of tht wll gurtee tht my pproxmto s wth error of E from the ctul re? Grded Exmple: Use the mdpot rule wth to pproxmte the vlue of Soluto: We see tht ths cse d. Ths mes tht mples tht x0, x 0, d x. So, f we let x 4 4 e dx f x f x f f e e 4.558. e x e dx. x. Ths e x y the mdpot rule we see: Trpezodl Rule The de here s to use trpezods rther th rectgles to pproxmte the re. The chllege s to pck the trpezods so tht ther res c e computed quckly d effcetly. The de (wth 4 ) of the trpezodl rule s show pctorlly elow: T T 3 T T 4 x 0 x x x 3 x 4 Note: We wll use the mes dscussg the mdpot rule. x d x the sme wy tht we used these mes whe
x4 I the ove pcture the pproxmte vlue of dx s gve y the sum of the re of the four trpezods. Now, we wll compute ths sum. Frst, recll tht the re of trpezod s gve y the formul: A heght of the trpezod d d see tht the re of T s 0 x0 h where h s the represet the legth of the ses. Usg ths formul we x f x f x. Followg ths ptter, we see tht the sum of the res of ll four trpezods s: x x x x f x0 f x f x f x f x f x3 f x3 f x4. If we x fctor out d come lke terms we see tht ths expresso s equl to: x f x 0 f x f x f x 3 f x 4. Geerlzg ths for rtrry leds to the trpezodl rule. x The Trpezodl Rule: dx f x f x f x... f x f x where x d x x for 0,,...,. 0 Grded Exmple: Use the trpezodl rule wth to pproxmte the vlue of Soluto: We see tht ths cse d. Ths mes tht mples tht x0, x 0, d x. So, f we let x e dx. x. Ths e x y the trpezodl rule we see: x e dx f f (0) f () e e e.367. Questo to the Clss: Whch do you thk gves the etter estmte the trpezodl or mdpot rule? Possle Aswer: Bsed upo the grded exmples we c tt some ecdotl evdece. If we trust tht the clcultor c fd x e dx to hgh level of ccurcy (we get e x dx.494 from the clcultor), we see tht the mout the mdpot rule ws off (for the grded exmple) ws pproxmtely:.558.494.064 d the mout the trpezodl rule ws off ws pproxmtely:.367.494.7. So, ths exmple the mdpot rule ppered to do etter jo.
Theorem: Suppose K Trpezodl d Mdpot rules, the for ll x,. If E T d 3 3 K K ET d E M. 4 E M re the errors the Proof: The proof of ths theorem wll e wthheld. We wll e le to prove ths theorem qute esly fter we fsh chpter. Remrk: From ths theorem we see tht we hve etter upper oud o the error from the Mdpot rule. So, ths sese the mdpot rule s etter. However, there re some exmples whch the trpezodl rule wll do just s well s the mdpot rule of pproxmtg the vlue of defte tegrl. It s mportt to uderstd tht the ove theorem s ot gvg you the error; rther, t s gvg you upper oud o the error ( ctulty the error mght e much smller th the upper oud). Note: Iterestgly, ppled mthemtcs (mely umercl lysts) would vew the Mdpot d Trpezodl rules s roughly the sme wth respect to how good they re sce oth errors c e oud y costt tmes. Exmple: () Suppose we pproxmte x e dx wth the mdpot rule d 5 upper oud for the error volved ths pproxmto. From the theorem we kow we must fd K so tht K for every x. Gve, ( ths cse e x x e 4x. Now, usg the closed tervl method (o pge 78) from Clc we c fd the glol mx d glol m of,. After dog ths we see tht f( x). So, we c tke K. ). We ote tht o Now, from the theorem we see tht upper oud for the error s: 3 6.06. 4 5 600 75 () How lrge should we tke f we wt to use the mdpot rule d ot x pproxmto of e dx tht s wth 000 of the ctul vlue of x e dx?
Smpso s Rule 6 The desred s such tht. Solvg ths equlty for > 0 we 4 000 6 000 000 hve: 6000 4. The sce 4 000 3 3 000 6, we kow we c tke 6 to cheve the desred result. 3 The de here s to use prols sted of strght le segmets (we use strght le segmets for the trpezodl rule) to pproxmte curve. We wll use the mes x d x the sme wy tht we used these mes whe dscussg the mdpot d trpezodl rules. Sce 3 ocoler pots determe prol we use the pots x, f x, x, f x, d x, f x for,4,6,..., to determe prol. The, to pproxmte the re uder the curve, we fd the re ouded etwee the x-xs d the prol (from x to x ). Ths s llustrted elow: Note: It s mportt to uderstd tht the re ove yellow s smply oe secto of Smpso s rule. I ctulty there c e my pots the prtto of the x-xs. Sce y prol tht we fd Smpso s rule wll exst over two of the sutervls (whch, s dvded to) t must e the cse tht s eve whe we pply Smpso s rule.
Due to lck of lecture tme we wll ot derve Smpso s rule. If you would lke to see the dervto t s o pges 5 d 5 of the textook. We ow preset Smpso s rule d the oud o the error (wthout proof) whe Smpso s rule s ppled. Smpso s Rule: x dx f x0 4 f x f x 4 f x3... f x 4f x f x where s 3 eve d x. Theorem: Suppose Smpso s rule the: 4 K for ll x, E S. If 5 K. 4 80 E S s the error volved usg Remrk: Oe should ote tht the error oud for Smpso s rule hs the potetl to e much smller th the error ouds for the Trpezodl d Mdpot rules. Grded Exmple: () Use Smpso s rule wth 4 to pproxmte the vlue of (roud your swer to four decml plces). Soluto: We see tht ths cse d. Ths mes tht x. Ths 4 mples tht x0, x, x 0, x3, d x4. So, f we let e x y x Smpso s rule we see: e dx f x0 4 f x f x 4 f x3 f x4.4944. 6 () Suppose tht you kow tht f e x the e x 4 for y x,. Use ths fct to fd upper oud o the error of the pproxmto you foud (). Soluto: 4 5 By the theorem the error s ouded ove y 384.0083. 80 4 46080 0 Recommeded Homework: 7.7-3-9 odd; 37 dx