3.4 Numerical integration

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "3.4 Numerical integration"

Transcription

1 3.4. Numericl integrtion Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [, b], i.e. I( f )= w(x) f (x) dx. The weight function my be the identity function w(x) =1, so tht the integrl represents the re under the function f in the intervl. In other pplictions the weight function could lso be the probbility density function of continuous rndom vrible x with support [, b], sothti( f ), represents the expected vlue of f ( x). In the following we discuss so-clled numericl qudrture methods. In order to pproximte the bove integrl, qudrture methods choose discrete set of qudrture nodes x i nd pproprite weights w i in wy tht n I( f ) w i f (x i ). The integrl is therefore pproximted by the sum of function vlues t the nodes x 0 < < x n b nd the respective qudrture weights w i. Of course, the pproximtion improves with rising number n. Different qudrture methods only differ with respect to the chosen nodes x i nd weights w i. In the following we will concentrte on Newton-Cotes method nd Gussin qudrture method. The former pproximtes the integrnd f between nodes using low-order polynomils nd sum the integrls of the polynomils to estimte the integrl of f. The ltter methods choose the nodes nd weights in order to mtch specific moments(such s expected vlue, vrince etc.) of the pproximted function Summed Newton-Cotes methods In this section we set w(x) =1. A weighted integrl cn then simply be computed by setting g(x) =w(x) f (x) nd pproximting g(x) dx. Summed Newton-Cotes formuls prtition the intervl [, b] into n subintervls of equl length h by computing the qudrture nodes x i := + ih, i = 0, 1,..., n, mit h := b n. A summed Newton-Cotes formul of degree k then interpoltes the function f on every subintervl [x i, x i+1 ] with polynomil of degree k. Wefinlly integrte the interpolting polynomil on every sub-intervl nd sum up ll the resulting sub-res. Figure 3.10 shows this pproch for k = 0ndk = 1.

2 64 Chpter 3 Numericl Solution Methods f(x) f(x) f(x) f(x) = x 1 x x 3 x 4 x 5 b = x 6 x = x 1 x x 3 x 4 x 5 b = x 6 x Figure 3.10: Summed Newton-Cotes formuls for k = 0 nd k = 1 Summed rectngle rule If k = 0, the interpolting functions re polynomils of degree 0, i.e. constnt functions. We cn write ny re below this constnt functions on the intervl [x i, x i+1 ] s I (0) [x i,x i+1 ] ( f )=(x i+1 x i ) f (x i )=hf(x i ), which is the surfce of rectngle. The degree 0 formul is therefore clled the summed rectngle rule, the explicit form of which is given by I (0) n 1 ( f )= hf(x i ). The weights of the summed rectngle rule consequently re w i = h for i = 0,...,n 1 nd w n = 0. Summed trpezoid rule If k = 1, the function f in the i-th subintervl [x i, x i+1 ] is pproximted by the line segment pssing through the two points (x i, f (x i )) nd (x i+1, f (x i+1 )). The re under this line segment is the surfce of trpezoid, i.e. I (1) [x i,x i+1 ] ( f )=hf(x i)+ 1 h[ f (x i+1) f (x i )] = h [ f (x i)+ f (x i+1 )]. It is immeditely cler tht the summed trpezoid rule improves the pproximtion of I( f ) compred to the summed rectngle rule discussed bove. Summing up the res of the trpezoids cross sub-intervls yields the pproximtion { } I (1) = h n 1 f ()+ f (x i )+ f (b) i=1 of the whole integrl I( f ). The weights of the rule therefore re w 0 = w n = h nd w i = h for i = 1,...,n 1. (3.5)

3 3.4. Numericl integrtion 65 Summed rectngle nd trpezoid rule re simple nd robust. Hence, computtion does not need much effort. In ddition, the ccurcy of the pproximtion of I( f ) increses with rising n. It is lso cler tht the trpezoid rule will exctly compute the integrl of ny first-order polynomil, i.e. line. Progrm 3.11 shows how to pply the summed trpezoid rule to the function cos(x). We Progrm 3.11: Summed trpezoid rule with cos(x) progrm NewtonCotes! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: h, x(0:n), w(0:n), f(0:n) integer :: i! clculte qudrture nodes h = (b-)/dble(n) x = (/( + dble(i)*h,,n)/)! get weights w(0) = h/d0 w(n) = h/d0 w(1:n-1) = h! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)-sin(0d0) end progrm thereby first declre ll the vribles needed. Specil ttention should be devoted to the declrtion of x, w nd f. These rrys will store the qudrture nodes x i, the weights w i nd the function vlues t the nodes f (x i ), respectively. We then first compute h nd the nodes x i = + ih nd the weights w i s in (3.5). With the respective function vlues, the pproximtion to the integrl 0 cos(x) dx = sin() sin(0) is the given by sum(w*f, 1). Note tht the pproximtion of the integrl is quite bd. If we increse the number of qudrture nodes n increses the ccurcy, however, we need bout 600 nodes in order to perfectly mtch numericl nd nlyticl result on 6 digits.

4 66 Chpter 3 Numericl Solution Methods Summed Simpson rule Finlly if k =, the integrnd f in the i-th subintervl [x i, x i+1 ] is pproximted by second-order polynomil function c 0 + c 1 x + c x.nowthreegrph points re required to specify the polynomil prmeters c i in the subintervls. Given these prmeters, one cn compute the integrl of the qudrtic function in the subintervl [x i, x i+1 ] s [ ( ) ] I () [x i,x i+1 ] ( f )=h xi + x f (x i )+4f i+1 + f (x i+1 ). 3 Summing up the different res on the sub-intervls yields { I () ( f )= h n 1 n 1 ( ) } xi + x f ()+ 6 f (x i )+4 f i+1 + f (b). i=1 The summed Simpson rule is lmost s simple to implement s the trpezoid rule. If the integrnd is smooth, Simpson s rule yields pproximtion error tht with rising n flls twice s fst s tht of the trpezoid rule. For this reson Simpson s rule is usully preferred to the trpezoid nd rectngle rule. Note, however, tht the trpezoid rule will often be more ccurte thn Simpson s rule if the integrnd exhibits discontinuities in its first derivtive, which cn occur in economic pplictions exhibiting corner solutions. Of course, summed Newton-Cotes rules lso exist for higher order piecewise polynomil pproximtions, but they re more difficult to work with nd thus rrely used Gussin Qudrture In opposite to the Newton-Cotes methods, Gussin qudrture tkes explicitly into ccount weight functions w(x). Here obviously, qudrture nodes x i nd weights w i re computed differently. Guss-Legendre qudrture Suppose gin, for the moment, tht w(x) =1. The Guss-Legendre qudrture nodes x i [, b] nd weights w i re computed in wy tht they stisfy the n + momentmtching conditions x k w(x) dx = x k n dx = w i xi k for k = 0, 1,..., n 1. (3.6) With the bove conditions holding, we cn write every integrl over polynomil with degree m n 1s p(x) dx = c 0 p(x) =c 0 + c 1 x c m x m 1 dx +c 1 xdx } {{} = n w ix i }{{} = n w i c m x m dx } {{} = n w ixi m

5 3.4. Numericl integrtion 67 n = w i [c 0 + c 1 x i c m x m n i ] = w i p(x i ). Consequently, qudrture nodes nd weights tht stisfy the moment-mtching conditions in (3.6) re ble to integrte ny polynomil p of degree m n 1exctly. Clculting the nodes nd weights of the Guss-Legendre qudrture formul is not so esy. One method is to set up the non-liner eqution system defined in (3.6). For n = 3 e.g. we hve w 0 x 0 + w 1 x 1 + w x = 1 ) (b = 1 dx, w 0 x 0 + w 1 x 1 + w x = 1 ( b ) = xdx,. w 0 x0 5 + w 1x1 5 + w x 5 = 1 ( b 6 6) = x 5 dx. 6 This nonliner eqution system cn be solved by rootfinding lgorithm like fzero. However, there is more efficient, but lso less intuitive wy to clculte qudrture nodes nd weights of the Guss-Legendre qudrture implemented in the subroutine legendre in the module gussin_int. Progrm3.1 demonstrtes how to use it. The Progrm 3.1: Guss-Legendre qudrture with cos(x) progrm GussLegendre! module use use gussin_int! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: x(0:n), w(0:n), f(0:n)! clculte nodes nd weights cll legendre(0d0, d0, x, w)! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)-sin(0d0) end progrm subroutine tkes two rel*8 rguments defining the left nd right intervl borders nd

6 68 Chpter 3 Numericl Solution Methods b. In ddition, we hve to pss two rrys of equl length to the routine. The first of these will be filled with the nodes x i, wheres the second will be given the weights w i. After hving clculted the function vlues f (x i ), we cn gin clculte the numericl pproximtion of the integrl like in Progrm Here, with 10 qudrture nodes, we lredy mtch the nlyticl integrl vlue by 6 digits. Note tht we needed bout 600 nodes with the trpezoid rule. Guss-Hermite qudrture If we set the weight function to w(x) = 1 π exp( x ),weresultintheguss-hermite qudrture method. Note tht the weight function now is equl to the density function of the stndrd norml distribution. Now, with m n + 1 nd normlly distributed rndom vrible x, wehve E( x) = 1 exp( x )x m n dx = π w i xi m = I GH (x m ). Consequently, with Guss-Hermite qudrture, we re ble to perfectly mtch the first n 1 moments of normlly distributed rndom vrible x. An pproximtion procedure for normlly distributed rndom vribles is included in the module normlprob. The procedure norml_discrete(x, w, mu, sig) is exctly bsed on the Guss-Hermite qudrture method. The subroutine receives four input rguments, the lst of which re expected vlue nd vrince of the norml distribution tht should be pproximted. The routine then stores pproprite nodes x i nd weights w i in the rrys x nd w (tht should be of sme length) which cn be used to compute moments of normlly distributed rndom vrible x with expecttion mu nd vrince sig. For pplying this subroutine we consider the following exmple: Exmple Consider n griculturl commodity mrket, where plnting decisions re bsed on the price expected t hrvest A = E(p), (3.7) with A denoting crege supply nd E(p) defining expected price. After the crege is plnted, normlly distributed rndom yield y N(1, 0.1) is relized, giving rise to the quntity q s = Ay which is sold t the mrket clering price p = 3 q s. In order to solve this system we substitute nd therefore q s = [ E(p)] y p = 3 [ E(p)] y.

7 3.4. Numericl integrtion 69 Tking expecttions on both sides leds to E(p) =3 [ E(p)] E(y) nd therefore E(p) = 1. Consequently, equilibrium crege is A = 1. Finlly, the equilibrium price distribution hs vrince of Vr(p) =4 [ E(p)] Vr(y) =4Vr(y) =0.4. Suppose now tht the government introduces price support progrm which gurntees ech producer minimum price of 1. If the mrket price flls below this level, the government pys the producer the difference per unit produced. Consequently, the producer now receives n effective price of mx(p,1) nd the expected price in (3.7) thenis clculted vi E(p) =E [mx (3 Ay,1)]. (3.8) The equilibrium crege supply finlly is the supply A tht fulfills (3.7) with the bove price expecttion. Agin, this problem cnnot be solved nlyticlly. In order to compute the solution of the bove crege problem, one hs to use two modules. normlprob on the one hnd provides the method to discretize the normlly distributed rndom vrible y. The solution to (3.7) cn on the other hnd be clculted by using the method fzero from the rootfinding module. Progrm 3.13 shows how to do this. Due to spce restrictions, we do not show the whole progrm. For running the progrm, we lso need module globls in which we store the expecttion mu nd vrince sig of the normlly distributed rndom vrible y s well s the minimum price minp gurnteed by the government. In ddition, the module stores the qudrture nodes y nd weights w obtined by discretizing the norml distribution for y. In the progrm, we first discretize the distribution for y by mens of the subroutine norml_discrete. This routine receives the expected vlue nd vrince of the norml distribution nd stores the respective nodes y nd weights w in two rrys of sme size. Next we set strting guess for crege supply. We then let fzero find the root of the function mrket tht clcultes the mrket equilibrium condition in (3.7). This function only gets A s n input. From A we cn clculte the expected price E(p) by mens of (3.8) nd finlly the mrket clering condition. Hving found the root of mrket, i.e. the equilibrium crege supply, we cn clculte the expected vlue nd vrince of the price p. Inorder to showthe effects of minimum price, wefirst set the minimum price gurntee minp to lrge negtive vlue, sy 100. This lrge negtive minimum price will never be binding, hence, the outcome is exctly the sme s in the model without price gurntee, see the bove exmple description. If we now set the minimum price t 1, equilibrium crege supply increses by bout 9.7percent. This is becuse the expected effective price

8 70 Chpter 3 Numericl Solution Methods Progrm 3.13: Agriculturl problem progrm griculture...! discretize y cll norml_discrete(y, w, mu, sig)! initilize vribles A = 1d0! get optimum cll fzero(a, mrket, check)! get expecttion nd vrince of price Ep = sum(w*mx(3d0-d0*a*y, minp)) Vrp = sum(w*(mx(3d0-d0*a*y, minp) - Ep)**)... end progrm function mrket(a) use globls rel*8, intent(in) :: A rel*8 :: mrket rel*8 :: Ep! clculte expected price Ep = sum(w*mx(3d0-d0*a*y, minp))! get equilibrium eqution mrket = A - (0.5d0+0.5d0*Ep) end function for the producer rises from the minimum price gurntee. In ddition, the uncertinty for the producers is reduced, since the vrince of the price distribution decreses from 0.4 to bout 0.115, which is gin due to the fct tht the possible price reliztion now hve lower limit of 1.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

Numerical integration

Numerical integration 2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Numerical Integration

Numerical Integration Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the

More information

CMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature

CMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy

More information

Definite integral. Mathematics FRDIS MENDELU

Definite integral. Mathematics FRDIS MENDELU Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

More information

Numerical Integration

Numerical Integration Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture

More information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30 Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

Theoretical foundations of Gaussian quadrature

Theoretical foundations of Gaussian quadrature Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

More information

DOING PHYSICS WITH MATLAB MATHEMATICAL ROUTINES

DOING PHYSICS WITH MATLAB MATHEMATICAL ROUTINES DOIG PHYSICS WITH MATLAB MATHEMATICAL ROUTIES COMPUTATIO OF OE-DIMESIOAL ITEGRALS In Cooper School of Physics, University of Sydney in.cooper@sydney.edu.u DOWLOAD DIRECTORY FOR MATLAB SCRIPTS mth_integrtion_1d.m

More information

Construction of Gauss Quadrature Rules

Construction of Gauss Quadrature Rules Jim Lmbers MAT 772 Fll Semester 2010-11 Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht Newton-Cotes qudrture

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but... Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.

More information

Numerical quadrature based on interpolating functions: A MATLAB implementation

Numerical quadrature based on interpolating functions: A MATLAB implementation SEMINAR REPORT Numericl qudrture bsed on interpolting functions: A MATLAB implementtion by Venkt Ayylsomyjul A seminr report submitted in prtil fulfillment for the degree of Mster of Science (M.Sc) in

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Numerical Integration. 1 Introduction. 2 Midpoint Rule, Trapezoid Rule, Simpson Rule. AMSC/CMSC 460/466 T. von Petersdorff 1

Numerical Integration. 1 Introduction. 2 Midpoint Rule, Trapezoid Rule, Simpson Rule. AMSC/CMSC 460/466 T. von Petersdorff 1 AMSC/CMSC 46/466 T. von Petersdorff 1 umericl Integrtion 1 Introduction We wnt to pproximte the integrl I := f xdx where we re given, b nd the function f s subroutine. We evlute f t points x 1,...,x n

More information

Orthogonal Polynomials

Orthogonal Polynomials Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils

More information

III. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9

III. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9 III Lecture on Numericl Integrtion File fclib/dttb/lecture-notes/numerical-inter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the

More information

LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and

LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only

More information

Lecture 14 Numerical integration: advanced topics

Lecture 14 Numerical integration: advanced topics Lecture 14 Numericl integrtion: dvnced topics Weinn E 1,2 nd Tiejun Li 2 1 Deprtment of Mthemtics, Princeton University, weinn@princeton.edu 2 School of Mthemticl Sciences, Peking University, tieli@pku.edu.cn

More information

Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014

Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014 Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method

More information

1 The Lagrange interpolation formula

1 The Lagrange interpolation formula Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x

More information

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004 Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

More information

Abstract inner product spaces

Abstract inner product spaces WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

More information

Best Approximation. Chapter The General Case

Best Approximation. Chapter The General Case Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

More information

NUMERICAL INTEGRATION

NUMERICAL INTEGRATION NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls

More information

COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature

COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III - Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting

More information

The Riemann Integral

The Riemann Integral Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

Lecture 20: Numerical Integration III

Lecture 20: Numerical Integration III cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Chapter 3 Solving Nonlinear Equations

Chapter 3 Solving Nonlinear Equations Chpter 3 Solving Nonliner Equtions 3.1 Introduction The nonliner function of unknown vrible x is in the form of where n could be non-integer. Root is the numericl vlue of x tht stisfies f ( x) 0. Grphiclly,

More information

1 Probability Density Functions

1 Probability Density Functions Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1 Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

More information

Math 426: Probability Final Exam Practice

Math 426: Probability Final Exam Practice Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by

More information

Numerical Analysis: Trapezoidal and Simpson s Rule

Numerical Analysis: Trapezoidal and Simpson s Rule nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =

More information

Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature

Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

APPROXIMATE INTEGRATION

APPROXIMATE INTEGRATION APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 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 e determined

More information

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS. THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

Chapter 5 : Continuous Random Variables

Chapter 5 : Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll

More information

Overview of Calculus I

Overview of Calculus I Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

More information

n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1 The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

Lecture 23: Interpolatory Quadrature

Lecture 23: Interpolatory Quadrature Lecture 3: Interpoltory Qudrture. Qudrture. The computtion of continuous lest squres pproximtions to f C[, b] required evlutions of the inner product f, φ j = fxφ jx dx, where φ j is polynomil bsis function

More information

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s). Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

More information

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve. Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

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

c n φ n (x), 0 < x < L, (1) n=1

c n φ n (x), 0 < x < L, (1) n=1 SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

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

Chapter 5. Numerical Integration

Chapter 5. Numerical Integration Chpter 5. Numericl Integrtion These re just summries of the lecture notes, nd few detils re included. Most of wht we include here is to be found in more detil in Anton. 5. Remrk. There re two topics with

More information

COT4501 Spring Homework VII

COT4501 Spring Homework VII COT451 Spring 1 Homework VII The ssignment is due in clss on Thursdy, April 19, 1. There re five regulr problems nd one computer problem (using MATLAB). For written problems, you need to show your work

More information

Review of basic calculus

Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Least Squares Approximation Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

More information

Numerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden

Numerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden Numericl Anlysis 10th ed R L Burden, J D Fires, nd A M Burden Bemer Presenttion Slides Prepred by Dr. Annette M. Burden Youngstown Stte University July 9, 2015 Chpter 4.1: Numericl Differentition 1 Three-Point

More information

Numerical Methods I Orthogonal Polynomials

Numerical Methods I Orthogonal Polynomials Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX

More information

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

More information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), ) Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s

More information

Advanced Computational Fluid Dynamics AA215A Lecture 3 Polynomial Interpolation: Numerical Differentiation and Integration.

Advanced Computational Fluid Dynamics AA215A Lecture 3 Polynomial Interpolation: Numerical Differentiation and Integration. Advnced Computtionl Fluid Dynmics AA215A Lecture 3 Polynomil Interpoltion: Numericl Differentition nd Integrtion Antony Jmeson Winter Qurter, 2016, Stnford, CA Lst revised on Jnury 7, 2016 Contents 3 Polynomil

More information

Numerical Integration. Newton Cotes Formulas. Quadrature. Newton Cotes Formulas. To approximate the integral b

Numerical Integration. Newton Cotes Formulas. Quadrature. Newton Cotes Formulas. To approximate the integral b Numericl Integrtion Newton Cotes Formuls Given function f : R R nd two rel numbers, b R, < b, we clculte (pproximtely) the integrl I(f,, b) = f (x) dx K. Frischmuth (IfM UR) Numerics for CSE 08/09 8 /

More information

Improper Integrals, and Differential Equations

Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Carlo Integration 02/10/05 CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This

More information

UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3

UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,

More information

SYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus

SYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is

More information

Orthogonal Polynomials and Least-Squares Approximations to Functions

Orthogonal Polynomials and Least-Squares Approximations to Functions Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

More information

1 Part II: Numerical Integration

1 Part II: Numerical Integration Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble

More information

JDEP 384H: Numerical Methods in Business

JDEP 384H: Numerical Methods in Business BT 3.4: Solving Nonliner Systems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods Instructor: Thoms Shores Deprtment of Mthemtics Lecture 20, Februry 29, 2007 110 Kufmnn Center Instructor:

More information

AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION

AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION

More information

f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all

f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all 3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

The goal of this section is to learn how to use a computer to approximate definite integrals, i.e. expressions of the form. Z b

The goal of this section is to learn how to use a computer to approximate definite integrals, i.e. expressions of the form. Z b Lecture notes for Numericl Anlysis Integrtion Topics:. Problem sttement nd motivtion 2. First pproches: Riemnn sums 3. A slightly more dvnced pproch: the Trpezoid rule 4. Tylor series (the most importnt

More information

MA 124 January 18, Derivatives are. Integrals are.

MA 124 January 18, Derivatives are. Integrals are. MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,

More information

1 The Riemann Integral

1 The Riemann Integral The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

7 - Continuous random variables

7 - Continuous random variables 7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

More information

THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS

THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2-Dimensionl Gussin Qudrture 20 4

More information

1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),

1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ), 1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on

More information

Chapter 0. What is the Lebesgue integral about?

Chapter 0. What is the Lebesgue integral about? Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

More information

Lecture Note 4: Numerical differentiation and integration. Xiaoqun Zhang Shanghai Jiao Tong University

Lecture Note 4: Numerical differentiation and integration. Xiaoqun Zhang Shanghai Jiao Tong University Lecture Note 4: Numericl differentition nd integrtion Xioqun Zng Sngi Jio Tong University Lst updted: November, 0 Numericl Anlysis. Numericl differentition.. Introduction Find n pproximtion of f (x 0 ),

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Monte Carlo method in solving numerical integration and differential equation

Monte Carlo method in solving numerical integration and differential equation Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The

More information