the midpoint of the ith subinterval, and n is even for

Size: px
Start display at page:

Download "the midpoint of the ith subinterval, and n is even for"

Transcription

1 Mth4 Project I (TI 89) 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

2 The ove Approimtig Sums c e foud usig progrm o the clcultor clled riem. riem( ) Prgm Locl md,,,,h,s,c,l,r,t,m,i getmode( Ect/Appro ) md setmode( Ect/Appro, APPROXIMATE ) setmode( Disply Digits, FLOAT ) ClrIO Prompt,, ClrIO Output,,,,N Output,4, Output, 8, Output,, (-)/ h s For i,,- +h*i c s+y(c) s EdFor (y()+s)*h L (s+y())*h r (L+r)/ t Output,, L Output, 4, L Output,, R Output, 4, r Output 4,, T Output 4, 4, t m +h/c For i,, m+y(c) m c+h c EdFor m*h m Output 5,, M Output 5, 4, m Output 6,, S(N) Output 6, 4, (t+*m)/ setmode( Ect/Appro, md) EdPrgm To crete this progrm o your clcultor: Go to the Progrm Editor (fter pressig APPS). Press to crete ew progrm. Go dow to Vrile d type riem. Whe doe press ENTER twice. The you type the followig progrm etwee Prgm d EdPrgm. The ALPHA key lets you eter the lphetic chrcters. To eter r, for emple, press ALPHA d the press. If you hve to eter severl lphetic chrcters, press d ALPHA to get the ALOCK so tht you void pressig the ALPHA key my times. To get the cpitl letters press ALPHA To hve setmode( Ect/Appro, APPROXIMATE ), press d F to get F6, press D to get Ect/Appro, press ENTER to get APPROXIMATE To hve getmode( Ect/Appro ), modify the previous istructios y chgig s to g d delete APPROXIMATE To hve setmode( Disply Digits, FLOAT ), press d F scroll dow to E, press ENTER. The commds Prompt, Output re i F the progrm iput/output meu. The commds For d EdFor re i F, the commd Locl is i F4 The commds getmode, setmode, ClrIO re i CATALOG is d d is the STO key. To isert ew lie press ENTER or d ENTER Whe doe typig the progrm, Press HOME to leve the progrm editor. We c use the riem progrm to pproimte the followig itegrl: Press Y= d set y = ^. Press HOME to go to the home scree. Type riem( ) ENTER to eecute the progrm. The progrm will prompt you for, the left edpoit of the itervl,, the right edpoit of the itervl, d, the umer of prtitio. 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. d

3 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 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 = =.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 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.

4 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 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 F i your clcultor 4 ( /,,,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 =

5 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= s( / ) 5 The mimum vlue of f () is K =. Or use the followig iequlities: So f ( ) K (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)?

6 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)

Math1242 Project I (TI 84) Name:

Math1242 Project I (TI 84) Name: 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

More information

Approximate Integration

Approximate Integration Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:

More information

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right: Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the

More information

f ( x) ( ) dx =

f ( x) ( ) dx = Defiite Itegrls & Numeric Itegrtio Show ll work. Clcultor permitted o, 6,, d Multiple Choice. (Clcultor Permitted) If the midpoits of equl-width rectgles is used to pproximte the re eclosed etwee the x-xis

More information

Limit of a function:

Limit of a function: - Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive

More information

AP Calculus Notes: Unit 6 Definite Integrals. Syllabus Objective: 3.4 The student will approximate a definite integral using rectangles.

AP Calculus Notes: Unit 6 Definite Integrals. Syllabus Objective: 3.4 The student will approximate a definite integral using rectangles. AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojective:.4 The studet will pproimte defiite itegrl usig rectgles. Recll: If cr is trvelig t costt rte (cruise cotrol), the its distce trveled is equl to rte

More information

Definite Integral. The Left and Right Sums

Definite Integral. The Left and Right Sums Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give

More information

B. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i

B. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio

More information

Approximations of Definite Integrals

Approximations of Definite Integrals 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

More information

1 Tangent Line Problem

1 Tangent Line Problem October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,

More information

1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2

1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2 Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit

More information

Feedback & Assessment of Your Success. 1 Calculus AP U5 Integration (AP) Name: Antiderivatives & Indefinite Integration (AP) Journal #1 3days

Feedback & Assessment of Your Success. 1 Calculus AP U5 Integration (AP) Name: Antiderivatives & Indefinite Integration (AP) Journal #1 3days Clculus AP U5 Itegrtio (AP) Nme: Big ide Clculus is etire rch of mthemtics. Clculus is uilt o two mjor complemetry ides. The first is differetil clculus, which is cocered with the istteous rte of chge.

More information

Multiplicative Versions of Infinitesimal Calculus

Multiplicative Versions of Infinitesimal Calculus Multiplictive Versios o Iiitesiml Clculus Wht hppes whe you replce the summtio o stdrd itegrl clculus with multiplictio? Compre the revited deiitio o stdrd itegrl D å ( ) lim ( ) D i With ( ) lim ( ) D

More information

We saw in Section 5.1 that a limit of the form. 2 DEFINITION OF A DEFINITE INTEGRAL If f is a function defined for a x b,

We saw in Section 5.1 that a limit of the form. 2 DEFINITION OF A DEFINITE INTEGRAL If f is a function defined for a x b, 3 6 6 CHAPTER 5 INTEGRALS CAS 5. Fid the ect re uder the cosie curve cos from to, where. (Use computer lger sstem oth to evlute the sum d compute the it.) I prticulr, wht is the re if? 6. () Let A e the

More information

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =

More information

 n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!

 n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2! mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges

More information

4. When is the particle speeding up? Why? 5. When is the particle slowing down? Why?

4. When is the particle speeding up? Why? 5. When is the particle slowing down? Why? AB CALCULUS: 5.3 Positio vs Distce Velocity vs. Speed Accelertio All the questios which follow refer to the grph t the right.. Whe is the prticle movig t costt speed?. Whe is the prticle movig to the right?

More information

5.1 - Areas and Distances

5.1 - Areas and Distances Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 63u Office Hours: R 9:5 - :5m The midterm will cover the sectios for which you hve received homework d feedbck Sectios.9-6.5 i your book.

More information

1.3 Continuous Functions and Riemann Sums

1.3 Continuous Functions and Riemann Sums mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios

More information

REVIEW OF CHAPTER 5 MATH 114 (SECTION C1): ELEMENTARY CALCULUS

REVIEW OF CHAPTER 5 MATH 114 (SECTION C1): ELEMENTARY CALCULUS REVIEW OF CHAPTER 5 MATH 4 (SECTION C): EEMENTARY CACUUS.. Are.. Are d Estimtig with Fiite Sums Emple. Approimte the re of the shded regio R tht is lies ove the -is, elow the grph of =, d etwee the verticl

More information

Linford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)

Linford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4) Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem

More information

1.1 The FTC and Riemann Sums. An Application of Definite Integrals: Net Distance Travelled

1.1 The FTC and Riemann Sums. An Application of Definite Integrals: Net Distance Travelled mth 3 more o the fudmetl theorem of clculus The FTC d Riem Sums A Applictio of Defiite Itegrls: Net Distce Trvelled I the ext few sectios (d the ext few chpters) we will see severl importt pplictios of

More information

Fall 2004 Math Integrals 6.1 Sigma Notation Mon, 15/Nov c 2004, Art Belmonte

Fall 2004 Math Integrals 6.1 Sigma Notation Mon, 15/Nov c 2004, Art Belmonte Fll Mth 6 Itegrls 6. Sigm Nottio Mo, /Nov c, Art Belmote Summr Sigm ottio For itegers m d rel umbers m, m+,...,, we write k = m + m+ + +. k=m The left-hd side is shorthd for the fiite sum o right. The

More information

Crushed Notes on MATH132: Calculus

Crushed Notes on MATH132: Calculus Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify

More information

Convergence rates of approximate sums of Riemann integrals

Convergence rates of approximate sums of Riemann integrals Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem

More information

10. 3 The Integral and Comparison Test, Estimating Sums

10. 3 The Integral and Comparison Test, Estimating Sums 0. The Itegrl d Comriso Test, Estimtig Sums I geerl, it is hrd to fid the ect sum of series. We were le to ccomlish this for geometric series d for telescoig series sice i ech of those cses we could fid

More information

Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1

Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1 Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule

More information

n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1

n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1 Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3

More information

Numerical Integration

Numerical Integration Numericl tegrtio Newto-Cotes Numericl tegrtio Scheme Replce complicted uctio or tulted dt with some pproimtig uctio tht is esy to itegrte d d 3-7 Roerto Muscedere The itegrtio o some uctios c e very esy

More information

( ) = A n + B ( ) + Bn

( ) = A n + B ( ) + Bn MATH 080 Test 3-SOLUTIONS Fll 04. Determie if the series is coverget or diverget. If it is coverget, fid its sum.. (7 poits) = + 3 + This is coverget geometric series where r = d

More information

For students entering Honors Precalculus Summer Packet

For students entering Honors Precalculus Summer Packet Hoors PreClculus Summer Review For studets eterig Hoors Preclculus Summer Pcket The prolems i this pcket re desiged to help ou review topics from previous mthemtics courses tht re importt to our success

More information

5.3. The Definite Integral. Limits of Riemann Sums

5.3. The Definite Integral. Limits of Riemann Sums . The Defiite Itegrl 4. The Defiite Itegrl I Sectio. we ivestigted the limit of fiite sum for fuctio defied over closed itervl [, ] usig suitervls of equl width (or legth), s - d>. I this sectio we cosider

More information

The Definite Riemann Integral

The Definite Riemann Integral These otes closely follow the presettio of the mteril give i Jmes Stewrt s textook Clculus, Cocepts d Cotexts (d editio). These otes re iteded primrily for i-clss presettio d should ot e regrded s sustitute

More information

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING

THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier

More information

The Reimann Integral is a formal limit definition of a definite integral

The Reimann Integral is a formal limit definition of a definite integral MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl

More information

Theorem 5.3 (Continued) The Fundamental Theorem of Calculus, Part 2: ab,, then. f x dx F x F b F a. a a. f x dx= f x x

Theorem 5.3 (Continued) The Fundamental Theorem of Calculus, Part 2: ab,, then. f x dx F x F b F a. a a. f x dx= f x x Chpter 6 Applictios Itegrtio Sectio 6. Regio Betwee Curves Recll: Theorem 5.3 (Cotiued) The Fudmetl Theorem of Clculus, Prt :,,, the If f is cotiuous t ever poit of [ ] d F is tiderivtive of f o [ ] (

More information

Chapter 5 The Definite Integral

Chapter 5 The Definite Integral Sectio. Chpter The Defiite Itegrl Sectio. Estimtig with Fiite Sums (pp. -) Eplortio Which RAM is the Biggest?.. LRAM > MRAM > RRAM, ecuse the heights of the rectgles decrese s ou move towrd the right uder

More information

Important Facts You Need To Know/Review:

Important Facts You Need To Know/Review: Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t

More information

denominator, think trig! Memorize the following two formulas; you will use them often!

denominator, think trig! Memorize the following two formulas; you will use them often! 7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges

More information

In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is

In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I

More information

General properties of definite integrals

General properties of definite integrals Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties

More information

Calculus II Homework: The Integral Test and Estimation of Sums Page 1

Calculus II Homework: The Integral Test and Estimation of Sums Page 1 Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the

More information

AP Calculus AB AP Review

AP Calculus AB AP Review AP Clulus AB Chpters. Re limit vlues from grphsleft-h Limits Right H Limits Uerst tht f() vlues eist ut tht the limit t oes ot hve to.. Be le to ietify lel isotiuities from grphs. Do t forget out the 3-step

More information

Simpson s 1/3 rd Rule of Integration

Simpson s 1/3 rd Rule of Integration Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?

More information

Pre-Calculus - Chapter 3 Sections Notes

Pre-Calculus - Chapter 3 Sections Notes Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied

More information

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k. . Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric

More information

Riemann Integral and Bounded function. Ng Tze Beng

Riemann Integral and Bounded function. Ng Tze Beng Riem Itegrl d Bouded fuctio. Ng Tze Beg I geerlistio of re uder grph of fuctio, it is ormlly ssumed tht the fuctio uder cosidertio e ouded. For ouded fuctio, the rge of the fuctio is ouded d hece y suset

More information

Numerical Integration by using Straight Line Interpolation Formula

Numerical Integration by using Straight Line Interpolation Formula Glol Jourl of Pure d Applied Mthemtics. ISSN 0973-1768 Volume 13, Numer 6 (2017), pp. 2123-2132 Reserch Idi Pulictios http://www.ripulictio.com Numericl Itegrtio y usig Stright Lie Iterpoltio Formul Mhesh

More information

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before 8.1 Arc Legth Wht is the legth of curve? How c we pproximte it? We could do it followig the ptter we ve used efore Use sequece of icresigly short segmets to pproximte the curve: As the segmets get smller

More information

MA123, Chapter 9: Computing some integrals (pp )

MA123, Chapter 9: Computing some integrals (pp ) MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how

More information

Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,

More information

where i is the index of summation, a i is the ith term of the sum, and the upper and lower bounds of summation are n and 1.

where i is the index of summation, a i is the ith term of the sum, and the upper and lower bounds of summation are n and 1. Chpter Itegrtio. Are Use sigm ottio to write d evlute sum. Uderstd the cocept o re. Approimte the re o ple regio. Fid the re o ple regio usig limits. Sigm Nottio I the precedig sectio, ou studied tidieretitio.

More information

( ) dx ; f ( x ) is height and Δx is

( ) dx ; f ( x ) is height and Δx is Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio

More information

Review of the Riemann Integral

Review of the Riemann Integral Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of

More information

18.01 Calculus Jason Starr Fall 2005

18.01 Calculus Jason Starr Fall 2005 18.01 Clculus Jso Strr Lecture 14. October 14, 005 Homework. Problem Set 4 Prt II: Problem. Prctice Problems. Course Reder: 3B 1, 3B 3, 3B 4, 3B 5. 1. The problem of res. The ciet Greeks computed the res

More information

Logarithmic Scales: the most common example of these are ph, sound and earthquake intensity.

Logarithmic Scales: the most common example of these are ph, sound and earthquake intensity. Numercy Itroductio to Logrithms Logrithms re commoly credited to Scottish mthemtici med Joh Npier who costructed tle of vlues tht llowed multiplictios to e performed y dditio of the vlues from the tle.

More information

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)

More information

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: APPENDEX I. THE RAW ALGEBRA IN STATISTICS A I-1. THE INEQUALITY Exmple A I-1.1. Solve ech iequlity. Write the solutio i the itervl ottio..) 2 p - 6 p -8.) 2x- 3 < 5 Solutio:.). - 4 p -8 p³ 2 or pî[2, +

More information

EVALUATING DEFINITE INTEGRALS

EVALUATING DEFINITE INTEGRALS Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those

More information

INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)

INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS) Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

Math 131. Numerical Integration Larson Section 4.6

Math 131. Numerical Integration Larson Section 4.6 Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n

More information

REAL ANALYSIS STUDY MATERIAL. B.Sc. Mathematics VI SEMESTER CORE COURSE. (2011 Admission) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION

REAL ANALYSIS STUDY MATERIAL. B.Sc. Mathematics VI SEMESTER CORE COURSE. (2011 Admission) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION REAL ANALYSIS STUDY MATERIAL BSc Mthemtics VI SEMESTER CORE COURSE ( Admissio) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION THENJIPALAM, CALICUT UNIVERSITY PO, MALAPPURAM, KERALA - 67 65 57 UNIVERSITY

More information

MTH 146 Class 16 Notes

MTH 146 Class 16 Notes MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si

More information

12.2 The Definite Integrals (5.2)

12.2 The Definite Integrals (5.2) 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

More information

Section 6.3: Geometric Sequences

Section 6.3: Geometric Sequences 40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.

More information

UNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)

UNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006) UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte

More information

Graphing Review Part 3: Polynomials

Graphing Review Part 3: Polynomials Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)

More information

Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral

Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ

More information

Numerical Integration - (4.3)

Numerical Integration - (4.3) Numericl Itegrtio - (.). Te Degree of Accurcy of Qudrture Formul: Te degree of ccurcy of qudrture formul Qf is te lrgest positive iteger suc tt x k dx Qx k, k,,,...,. Exmple fxdx 9 f f,,. Fid te degree

More information

MTH 146 Class 7 Notes

MTH 146 Class 7 Notes 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

More information

Limits and an Introduction to Calculus

Limits and an Introduction to Calculus Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit

More information

Convergence rates of approximate sums of Riemann integrals

Convergence rates of approximate sums of Riemann integrals Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki

More information

The Definite Integral

The Definite Integral The Defiite Itegrl A Riem sum R S (f) is pproximtio to the re uder fuctio f. The true re uder the fuctio is obtied by tkig the it of better d better pproximtios to the re uder f. Here is the forml defiitio,

More information

Section IV.6: The Master Method and Applications

Section IV.6: The Master Method and Applications Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio

More information

SOLUTION OF SYSTEM OF LINEAR EQUATIONS. Lecture 4: (a) Jacobi's method. method (general). (b) Gauss Seidel method.

SOLUTION OF SYSTEM OF LINEAR EQUATIONS. Lecture 4: (a) Jacobi's method. method (general). (b) Gauss Seidel method. SOLUTION OF SYSTEM OF LINEAR EQUATIONS Lecture 4: () Jcobi's method. method (geerl). (b) Guss Seidel method. Jcobi s Method: Crl Gustv Jcob Jcobi (804-85) gve idirect method for fidig the solutio of system

More information

EXERCISE a a a 5. + a 15 NEETIIT.COM

EXERCISE a a a 5. + a 15 NEETIIT.COM - Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()

More information

Math 3B Midterm Review

Math 3B Midterm Review Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht

More information

AP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review

AP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle

More information

BC Calculus Review Sheet

BC Calculus Review Sheet BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re

More information

Summer Math Requirement Algebra II Review For students entering Pre- Calculus Theory or Pre- Calculus Honors

Summer Math Requirement Algebra II Review For students entering Pre- Calculus Theory or Pre- Calculus Honors Suer Mth Requireet Algebr II Review For studets eterig Pre- Clculus Theory or Pre- Clculus Hoors The purpose of this pcket is to esure tht studets re prepred for the quick pce of Pre- Clculus. The Topics

More information

GRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.

GRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1. GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt

More information

Math 104: Final exam solutions

Math 104: Final exam solutions Mth 14: Fil exm solutios 1. Suppose tht (s ) is icresig sequece with coverget subsequece. Prove tht (s ) is coverget sequece. Aswer: Let the coverget subsequece be (s k ) tht coverges to limit s. The there

More information

INTEGRATION 5.1. Estimating with Finite Sums. Chapter. Area EXAMPLE 1. Approximating Area

INTEGRATION 5.1. Estimating with Finite Sums. Chapter. Area EXAMPLE 1. Approximating Area Chpter 5 INTEGRATION OVERVIEW Oe of the gret chievemets of clssicl geometr ws to oti formuls for the res d volumes of trigles, spheres, d coes. I this chpter we stud method to clculte the res d volumes

More information

ALGEBRA II CHAPTER 7 NOTES. Name

ALGEBRA II CHAPTER 7 NOTES. Name ALGEBRA II CHAPTER 7 NOTES Ne Algebr II 7. th Roots d Rtiol Expoets Tody I evlutig th roots of rel ubers usig both rdicl d rtiol expoet ottio. I successful tody whe I c evlute th roots. It is iportt for

More information

MAS221 Analysis, Semester 2 Exercises

MAS221 Analysis, Semester 2 Exercises MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)

More information

( x y ) x y. a b. a b. Chapter 2Properties of Exponents and Scientific Notation. x x. x y, Example: (x 2 )(x 4 ) = x 6.

( x y ) x y. a b. a b. Chapter 2Properties of Exponents and Scientific Notation. x x. x y, Example: (x 2 )(x 4 ) = x 6. Chpter Properties of Epoets d Scietific Nottio Epoet - A umer or symol, s i ( + y), plced to the right of d ove other umer, vrile, or epressio (clled the se), deotig the power to which the se is to e rised.

More information

PROGRESSIONS AND SERIES

PROGRESSIONS AND SERIES PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.

More information

f(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.

f(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a. Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ

More information

y udv uv y v du 7.1 INTEGRATION BY PARTS

y udv uv y v du 7.1 INTEGRATION BY PARTS 7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product

More information

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx), FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the

More information

Course 121, , Test III (JF Hilary Term)

Course 121, , Test III (JF Hilary Term) Course 2, 989 9, Test III (JF Hilry Term) Fridy 2d Februry 99, 3. 4.3pm Aswer y THREE questios. Let f: R R d g: R R be differetible fuctios o R. Stte the Product Rule d the Quotiet Rule for differetitig

More information

The Basic Properties of the Integral

The Basic Properties of the Integral The Bsic Properties of the Itegrl Whe we compute the derivtive of complicted fuctio, like x + six, we usully use differetitio rules, like d [f(x)+g(x)] d f(x)+ d g(x), to reduce the computtio dx dx dx

More information

is an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term

is an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic

More information

Laws of Integral Indices

Laws of Integral Indices A Lws of Itegrl Idices A. Positive Itegrl Idices I, is clled the se, is clled the idex lso clled the expoet. mes times.... Exmple Simplify 5 6 c Solutio 8 5 6 c 6 Exmple Simplify Solutio The results i

More information

b a 2 ((g(x))2 (f(x)) 2 dx

b a 2 ((g(x))2 (f(x)) 2 dx Clc II Fll 005 MATH Nme: T3 Istructios: Write swers to problems o seprte pper. You my NOT use clcultors or y electroic devices or otes of y kid. Ech st rred problem is extr credit d ech is worth 5 poits.

More information

Mean Value Theorem for Integrals If f(x) is a continuous function on [a,b] then there

Mean Value Theorem for Integrals If f(x) is a continuous function on [a,b] then there Itegrtio, Prt 5: The Me Vlue Theorem for Itegrls d the Fudmetl Theorem of Clculus, Prt The me vlue theorem for derivtives tells us tht if fuctio f() is sufficietly ice over the itervl [,] the t some poit

More information

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

More information

Options: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.

Options: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG. O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8

More information