Math1242 Project I (TI 84) Name:

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Mth4 Project I (TI 84) Nme: Riem Sums d Defiite Itegrls The re uder the grph of positive fuctio is give y the defiite itegrl of the fuctio. The defiite itegrl c e pproimted y the followig sums: Left Riem Sum: f ) d L [ f ( ) f ( )... f ( )] ( Right Riem Sum: f ) d R [ f ( ) f ( )... f ( )] ( Midpoit Rule: f ) d M [ f ( ) f ( )... f ( )] ( Trpezoidl Rule: f ( ) d T [ f ( ) f ( )... f ( ) f ( )] Simpso s Rule: f ( ) d S [ f ( ) 4 f ( ) f ( )... 4 f ( ) f ( )] Where, i i, i the midpoit of the ith suitervl, d is eve for Simpso s Rule. L R It turs out tht the Trpezoidl pproimtio T d Simpso s pproimtio T / M / S I ll of these methods we get more ccurte pproimtios whe we icrese the vlue of. The Error i usig pproimtio is: Error = Actul vlue of the itegrl - Approimtio = f ( ) d - Approimtio Error Bouds for Midpoit d Trpezoidl Rules: K( ) K( ) Suppose tht f ( ) K for. The E M d E T 4 Error Bouds for Simpso s Rules: K ( ) Suppose tht f ( ) K for. The E S 4 8 These Error Bouds re very useful to estimte the errors d the ccurcy of the pproimtios without hvig to fid the vlue of these pproimtios, especilly for lrge s. These Error Bouds re lso helpful i estimtig the umer of prtitios required to gurtee specific ccurcy whe pproimtig itegrl. 5

The ove Approimtig Sums c e foud usig simple progrm o the clcultor clled RIEMANN. PROGRAM : RIEMANN : Flot : ClrHome : Prompt A, B, N : (B-A)/N H : S : For ( I,, N- ) : A+I*H X : S+Y (X) S : Ed : (Y (A)+S)*H L : (S+Y (B))*H R : (L+R)/T : Output( 4,, L ) : Output( 4, 7, L) : Output( 5,, R ) : Output( 5, 7, R) : Output( 6,, T ) : Output( 6, 7, T) : M : A+H/X : For( I,, N) : M+ Y (X) M : X+H X : Ed : M*H M : Output( 7,, M ) : Output( 7, 7, M) : Output( 8,, S(N) ) : Output( 8, 7, (T+*M)/) To crete this progrm o your clcultor: Press PRGM to disply the PRGM NEW meu Press ENTER to select : Crete New. The Nme = prompt is displyed (d the A-LOCK is o with the TI 8 Plus), type i RIEMANN the me of the progrm The commds Flot d ClrHome re i the CATALOG press d d scroll dow. The ALPHA key lets you eter the lphetic chrcters. To eter R, for emple, press ALPHA, relese it, d the press. If you hve to eter severl lphetic chrcters, press d ALPHA to get the A-lock so tht you void pressig the ALPHA key my times. After typig the progrm s me, press ENTER. The progrm editor is displyed. The commds Disp, Prompt d Output re i the PRGM I/O( iput/output) meu. Press PRGM. The commds For d Ed re i the PRGM CTL (cotrol) meu. Press PRGM To get the Y, press VARS to disply the Y-VARS meu, select : Fuctio to disply the Y fuctios. is ALPHA +, : is ALPHA., is the STO key. Whe doe typig the progrm, Press d QUIT to leve the progrm editor d retur to the home scree. We c use the RIEMANN progrm to pproimte the followig itegrl: Press Y = d set Y = X. Press d QUIT to go to the home scree. Press PRGM to disply the PRGM EXEC meu. Select the progrm s me d press ENTER to eecute the progrm. The progrm will prompt you for A, the left edpoit of the itervl, B, the right edpoit of the itervl, d N, the umer of prtitios. It gives you the Left Riem Sum L, the right Riem Sum R, the midpoit pproimtio M, the Trpezoidl pproimtio T, d Simpso s pproimtio S. To chge A, B, or N, just press ENTER. The progrm eecutes other time d prompts you to eter ew vlues for A, B, d N. d

Prolem Use the RIEMANN progrm to pproimte d. Set Y = X, A =, B =, d N =. Get the followig: L 6.84, R 9.4, T 8.4, M 7.98, S 8. Note tht to fid S, you tke N = 5. To get etter pproimtio of the itegrl, you icrese the umer of prtitios N. Fill i the followig tle to pproimte the itegrl d : (swers correct to 6 deciml plces) N L N R N T N M N S N 6.84 9.4 8.4 7.98 8 4 8 Give tht d 8 d usig the tle ove, Which method gve the est pproimtios? Which vlue of N gve the est pproimtios? Which method(s) gve uderestimte of the itegrl? Which method(s) gve overestimte of the itegrl? The error i the ove pproimtios is ERROR = Actul vlue of itegrl Approimtio With N =, the error i the left Riem sum is E L = 8-6.84 =.6 Fill i the tle elow to fid the errors i the ove pproimtios (use 6 deciml plces) N E L E R E T E M E S.6 -.4 -.4. 4 8 Which method gve the lest errors? Which vlue of N gve the lest errors? Note tht the Trpezoidl d Midpoit Rules re much more ccurte th the edpoit pproimtios. The size of the error i the Midpoit Rule is out hlf the size of the error i the Trpezoidl Rule. Simpso s Approimtios re the most ccurte.

Prolem 4 I this prolem we will use the RIEMANN progrm to pproimte the vlue of = d Use your clcultor with Y = 4 / (+ X ) to fill i the followig tle. (Aswers to 6 deciml plces) N L N R N T N M N S N 4 8 6 Usig the tle, pproimtely, how my prtitios re eeded to pproimte to withi.5: whe usig the Midpoit Rule? whe usig the Trpezoidl Rule? whe usig the Simpso s Rule? Note tht your swers might ot e the smllest umer of prtitios tht will give you such precisio. I prolem, we ler how to fid etter estimtes of the umer of prtitios y usig the Error Boud Formuls. Prolem I this prolem we del with the Actul Errors = Actul vlue of itegrl Approimtios, d the Estimtes of Errors usig the Error Bouds give o the first pge of this project. Cosider the fuctio f ( ) d the itegrl d 4. (Give swers with 6 deciml plces) A) I this prt we fid the ctul vlue of the errors whe pproimtig 4 d. (i) Fid M =, T =, d S = (ii) You c evlute the itegrl 4 d usig MATH 9 i your clcultor 4 fit(/x, X,, 4) = or y hd d l 4 (iii) For =, fid the ctul error E M = 4 d M = the ctul error E T =, d the ctul error E S =

B) It is possile to estimte these Errors without fidig the pproimtios M, T, d S. I this prt we fid estimte of the errors usig the Error Bouds formuls. Error Bouds for Midpoit d Trpezoidl Rules: K( ) K( ) Suppose tht f ( ) K for. The E M d E T 4 Error Bouds for Simpso s Rules: K ( ) Suppose tht f ( ) K for. The E S 4 8 (i) Fid the followig derivtives of f ( ) : () f (), f ( ), f ( ), f ( ) (ii) To fid K, sketch the grph of y = f () o the itervl [, 4] y y pressig Y= MATH / ^ to get Y = s( / ) 5 The mimum vlue of f () is K =. Or use the followig iequlities: So f ( ) K 4 64. 64 (iii) With = prtitios d usig the ove formuls for Error Bouds, fid ( Show your work) K( ) (4 ) E M 4 4(), d E T (iv) Sketch the grph of y = f ( ) o the itervl [, 4] to fid K Upper Boud (or Mimum) of f ( ), K d E S (v) Are the Actul Errors foud i prt A) comptile with the Error Bouds i prt B)?

C) (i) Use the Error Boud formuls to fid the mimum possile error (i.e. upper oud for the error) i pproimtig 4 d with = 5 d usig the Trpezoidl rule. E T (ii) Use the Error Boud formuls to fid the mimum possile error i pproimtig with = usig the Simpso s rule. E S 4 d (iii) Usig your swers to prt (i) d (ii), the umer of prtitios eeded to pproimte 4 d correct to deciml plces is pproimtely: = with the Trpezoidl rule, d = with the Simpso s rule. D) Use the Error Boud formuls to fid how lrge do we hve to choose so tht the pproimtios T, M, d S to the itegrl 4 d re ccurte to withi.: K( ) (4 ) Trpezoidl rule: E T.. () > = (.) Midpoit rule: = (show work) Simpso s rule: = (show work)

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