Numerical Integration

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Numerical Integration"

Transcription

1 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 globl interpoltion polynomil or spline interpoltion, provides bsis for numericl integrtion techniques. Let the definite integrl under considertion be I{f} = f(x)dx where [, b] is finite closed intervl. In this chpter, we primrily consider pproximtions to I{f} tht re of the form n I n+ {f} = j f(x j ) where the qudrture nodes re given by, j= x < x < x <... < x n b nd the rel coefficients j re known s the qudrture coefficients. The nodes x j re pressigned nd often eqully spced. Numericl pproximtion of definite integrls is desirble in two cses:. closed form of I{f} is not esily obtined, or. the vilble closed form of I{f} is too complicted for efficient numericl evlution, s the following exmple clerly illustrtes: [ x dt + t 4 = 4 log x + x ] + e x x + + [ ] tn x + tn x. x + x In interpoltory qudrture, the only type we will be studying in this chpter in ddition to Romberg nd dptive qudrture, we pproximte the function f(x) by the interpolting polynomil (Lgrnge form) P n (x) = n f(x j )L j (x) j= where L j (x) = (x x ) (x j x ) (x x j ) (x j x j ) (x x j+) (x j x j+ ) (x x n) (x j x n ).

2 Chp. 5. Numericl Integrtion CS44 Clss Notes 7 Hence, in which I n+ {f} = j = P n (x)dx = n f(x j ) j= L j (x)dx = L j (x)dx, j =,,..., n. n j f(x j ) Note tht the qudrture coefficients j re completely determined by the end points nd b nd the interpoltion nodes x j, j =,,..., n. Correspondingly, the qudrture error or trunction error is given by E n+ {f} = I{f} I n+ {f} = in which e n (x) is the interpoltion error [f(x) P n (x)]dx = j= e n (x) = (x x )(x x ) (x x n ) f (n+) (x ) (n + )! e n (x)dx where x x (x) (x, x,..., x n, x). If f(x) is polynomil of degree t most n, then f (n+) (x) = nd hence E n+ {f} =. We will use this observtion to obtin some bsic interpoltory qudrture rules. 5. Method of Exct Mtching This cn lso be clled Method of Undetermined Coefficients. 5.. The Trpezoidl Rule The trpezoidl rule uses the function vlue t two points x nd x to compute the integrl of the function in the intervl [x, x ]. For x = nd x = h, the integrl is pproximted by the trpezoidl rule I{f} = f(x)dx I {f} = f + f where the qudrture coefficients nd need to be determined. Observe tht the trpezoidl rule is exct for polynomils of degree zero nd one, i.e., for constnt nd stright line. Therefore, it is exct for the following functions:. f(x) =. f(x) = x From () we see tht nd from () we hve dx = + = h, xdx = h = h. Using these equtions, we get the vlues of the coefficients, = = h/, nd the trpezoidl rule cn be written s I {f} = h (f + f ), where h = x x. Observe tht this is lso the re of the shded trpezoid in Fig.??. Consequently, I{f} = x where E {f} is the error in trpezoidl rule. x f(x)dx = I {f} + E {f} = h [f + f ] + E {f},

3 Chp. 5. Numericl Integrtion CS44 Clss Notes 7 p(x) f f(x) f x = x =h Figure 5.: Trpezoidl rule. Composite Trpezoidl Rule. From the bsic trpezoidl rule we cn construct qudrture rule to compute n integrl over the intervl [, b] by dividing the intervl into N equl subintervls nd using the bsic trpezoidl rule for ech subintervl. Suppose the intervl is divided into N subintervls of equl length using the nodes x, x,..., x N, where = + ih, for i =,..., N, nd h = b N is the size of ech subintervl. Assuming f( ) = f i, the composite trpezoidl rule is given s f(x)dx = h [f + f + f + + f N + f N ] + E T, where E T is the error in the composite trpezoidl rule. h h x = x x x 3 x N- x =b N Figure 5.: Composite trpezoidl rule for equidistnt points. 5.. Simpson s Rule Simpson s rule uses 3 interpoltion nodes nd qudrtic interpolting polynomil. The generl form is I 3 {f} = f + f + f

4 Chp. 5. Numericl Integrtion CS44 Clss Notes 73 Since the Simpson s rule is exct for polynomils of degree zero, one, nd two, it is exct for the following functions.. f(x) =,. f(x) = x, nd 3. f(x) = x. Thus, from () we hve from (), we get nd from (3) we obtin h h h These equtions yield the qudrture coefficients Hence, the Simpson s rule is given s nd I{f} = x x dx = + + = h, xdx = h + h =, x dx = h + h = 3 h3 = = 3 h, = 4 3 h. I 3 {f} = h 3 [f + 4f + f ], f(x)dx = I 3 {f} + E 3 {f} = h 3 [f + 4f + f ] + E 3 {f}, where h = x x = x x, nd E 3 {f} is the error in Simpson s rule. f f - f f(x) p(x) x =-h x = Figure 5.3: Simpson s rule. x =h

5 Chp. 5. Numericl Integrtion CS44 Clss Notes 74 Composite Simpson s Rule. Now, let us derive the composite Simpson s rule for computing integrl over the intervl [, b]. Suppose we hve N equl subintervls of width h, i.e., b = hn. We lso define N + eqully spced points x j = + jh, for j =,..., N in intervl [, b]. The ith subintervl hs the endpoints nd, nd the midpoint, for i =,..., N. Applying the bsic Simpson s rule over ech subintervl, we obtin the composite rule f(x)dx = [ h 3 (f + 4f + f ) + h 3 (f + 4f 3 + f 4 ) + + h 3 (f N + 4f N + f N ) ] + E S [ ] = h N N f + 4 f i + f i + f N + E S, 3 i= where E S is the error in the composite Simpson s rule. i= h h h h x = x x x 3 x 4 x 5 x 6 x x x =b N- N- N Figure 5.4: Composite Simpson s rule. 5. The Trunction Error 5.. The Trpezoidl Rule Recll tht I{f} = f(x)dx = h [f + f ] + E {f} Let us compute the error in the trpezoidl rule for the polynomils of degree,, nd, i.e., for f(x) =, x, x. E {} = E {x} = E {x } = dx h [ + ] =, xdx h [ + h] =, x dx h [ + h ] = 6 h3.

6 Chp. 5. Numericl Integrtion CS44 Clss Notes 75 p(x) f f(x) f h Figure 5.5: Trunction error in trpezoidl rule. Since the error is zero for polynomils of degree or less, the trpezoidl rule is sid to hve degree of precision =. By Tylor s formul, f(x) = f + xf + x! f + x3 3! f + = (stright line) + x! f + x3 3! f + Therefore, we hve the following expression for error { } x E {f(x)} = E {f + xf } + E f + x3 6 f + Since the error for stright line is zero, the first term on the right hnd side is zero, i.e., E {f + xf } =. Thus, E {f(x)} = + f E {x } + 6 f E {x 3 } + = h3 f h4 4 f = h3 f + O(h 4 ) The bove expression of the trunction error in the trpezoidl rule is known s n symptotic error estimte. Over here, the term O(h 4 ) hs the following mening: function q(h) = O(h α ) s h, (red s q(h) is of the order h α ), mens tht there exist constnts h nd K such tht 5.. Simpson s Rule Recll tht I{f} = q(h) Kh α, < h h. h f(x)dx = h 3 [f + 4f + f ] + E 3 {f} Let us compute the error for polynomils of degree 3 or less.

7 Chp. 5. Numericl Integrtion CS44 Clss Notes 76 f(x) p(x) f - f f -h h Figure 5.6: Trunction error in Simpson s rule. E 3 {} = E 3 {x} = E 3 {x } = E 3 {x 3 } = h h h h dx h [ ] =, 3 xdx h [ h + + h] =, 3 x dx h 3 [h + + h ] =, x 3 dx h 3 [ h3 + + h 3 ] =. We expected Simpson s rule to be exct for polynomils of degree or less. It turns out tht Simpson s rule is lso exct for cubics. For f(x) = x 4, E 3 {x 4 } = h x 4 dx h 3 [h4 + + h 4 ] = 4 5 h5. Therefore, Simpson s rule hs degree precision = 3. Using Tylor s formul, f(x) = f + xf + x! f + x3 3! f + x4 4! f (iv) + = (degree 3 polynomil) + x4 4 f (iv) + x5 f v + We hve shown bove tht Simpson s rule is exct for cubics. Thus, nd the symptotic error estimte is E 3 {f} = + 4 f (iv) E 3 {x 4 } + f v E 3{x 5 } + E 3 {f} = h5 9 f (iv) + O(h 7 ). We should mention here tht one cn obtin strict error estimtes for the bove two qudrture rules, rther thn just symptotic estimtes. A complete discussion of this question, however, is outside the scope of this book. We only stte the result s follows.

8 Chp. 5. Numericl Integrtion CS44 Clss Notes 77 Theorem 5. If n interpoltory qudrture formul hs degree of precision m, then its trunction error is given by E n+ {f} = f (m+) (z) (m + )! E n+{x m+ } where < z < b. Thus, for the trpezoidl rule m =, nd While for Simpson s rule, m = 3 nd E {f} = f (z) E {x } = h3! f (z), x < z < x. E 3 {f} = f (iv) (z) E 3 {x 4 } = h5 4! 9 f (iv) (z), x < z < x. The trunction error for the composite trpezoid rule is given by E T = h3 N f (z (b )h ) = f (z ), b = Nh, where x < z < x n. Similrly, the error for the composite Simpson s rule is given by [ ] E S = h5 f (iv) (y ) + f (iv) (y ) + + f (iv) (y N ) 9 where < y i <, for i =,,..., N. This simplifies to where < y < b. E S = h5 N 9 f (iv) (y (b )h4 ) = f (iv) (y ), 8 b = Nh 5.3 Spline Qudrture Consider the definite integrl I{f} = f(x)dx. If we pproximte f(x) vi spline interpoltion, such s the cubic interpoltory spline s(x) discussed in Chpter??, where = x < x < < x n = b, h i = +, then we obtin n interesting pproximtion of I{f} tht is given by n I sp {f} = i= xi+ s i (x)dx, where s i (x) is the cubic spline function for the ith subintervl [, + ], nd is given by s i (x) = σ i (+ x) 3 + σ ( ) i+ (x ) 3 fi+ + σ i+ h i (x ) h i h i h i ( ) fi + σ i h i (+ x) h i (see Chpter??). Let us define w = h i (x ), w = w = h i (+ x)

9 Chp. 5. Numericl Integrtion CS44 Clss Notes 78 Thus, s i (x) = s i (w) = h i σ i w 3 + h i σ i+ w 3 + w[f i+ σ i+ h i ] + w[f i σ i h i ] = wf i + wf i+ + h i [σ i+(w 3 w) + σ i ( w 3 w)], nd Moreover, observing tht xi+ s i (x)dx = h i w dw = s i (w)dw. w d w =, nd (w 3 w)dw = we obtin xi+ x ( w 3 w)d w = 4, s i (x)dx = h i [f i + f i+ ] h3 i 4 [σ i + σ i+ ]. Note tht this eqution cn be regrded s the trpezoidl rule plus correction term. This correction term is given by τ i = h3 4 [s ( ) + s (+ )], which indictes tht if f (x) is not too bdly behved, then In other words if f (x) is well behved, we hve τ i h3 i f (θ) E {f}, < θ < +. xi+ which cn be remrkbly close to the integrl s i (x)dx = h i [f i + f i+ ] + E {f} xi+ f(x)dx. 5.4 Richrdson s Extrpoltion In mny clcultions wht one would relly like to know is the limiting vlue of certin quntity F (h) s F. Needless to sy, the work required for computing F (h) increses shrply s h pproches zero. Furthermore, the effects of rounding errors set prcticl limit on how smll h cn be chosen. Usully, one hs some knowledge of how the trunction error [F () F (h)] behves s h. Let F (h) = F () + α h p + O(h r ) where r > p, nd α is n unknown. Compute F for two step lengths: h nd qh, (q > ) F (h) = F () + α h p + O(h r ) F (qh) = F () + α (qh) p + O(h r ). Multiplying the first eqution by q p, the second by, nd dding, we get p F (h) F (qh) = [q p ]F () + O(h r ),

10 Chp. 5. Numericl Integrtion CS44 Clss Notes 79 or F () = [ F (h) + ] F (h) F (qh) q p + O(h r ). This simple technique known s Richrdson s extrpoltion improves the symptotic error bound from O(h p ) to O(h r ). The repeted ppliction of Richrdson s extrpoltion to numericl integrtion is known s Romberg Integrtion. As n illustrtion, consider the composite trpezoidl rule on N pnels, ech of width h, In Appendix??, we show tht T (h) = h f + N j= f j + f N. I{f} I = T (h) + α h + α h 4 + α 3 h 6 + Using pnels of hlf the width, we get ( ) ( ) ( ) ( ) h h h 4 h 6 I = T + α + α + α Multiplying this eqution by /4 nd dding to the previous eqution, we obtin [ ( ) 3 h 4 I = T 4 ] T (h) 3 6 α h 4 or I = [ ( ) h 4T 3 [ ( ) h = T + 3 ] T (h) ( T ( h 4 α h 4 + O(h 6 ) (5.) ) )] T (h) + O(h 4 ) (5.) Suppose b = Nh, nd let us denote the composite formul with N pnels by T N, nd the formul with N pnels by T N. Thus, ( ) h T N = T (h), T N = T. The formul using Richrdson s extrpoltion is given by Now, eqution?? cn be written s Similrly in which T () N = 3 (4T N T N ) = T N + 3 (T N T N ). I = T () N 4 α h 4 + O(h 6 ). (5.3) I = T () 4N 4 α ( ) 4 h + O(h 6 ), (5.4) T () 4N = 3 (4T 4N T N ) = T 4N + 3 (T 4N T N ). Eliminting α from equtions (??) nd (??), we hve I = 5 = [ 6T () 4N T () N [ T () 4N + 5 = T () 4N + O(h6 ). ] + O(h 6 ) ( T () 4N T () N ) ] + O(h 6 )

11 Chp. 5. Numericl Integrtion CS44 Clss Notes 8 If we systemticlly hlve the intervl, tking h, h, h 4,..., etc., we cn construct the tble of which section is shown below. T N T () N T N T () 4N T 4N T () 4N T () 8N T () 8N T (3) 8N T 8N Error : O(h ) O(h 4 ) O(h 6 ) O(h 8 ) In generl, I = T (j) + O(h r ) in which r = (j + ), nd T (j) i = 4j T (j ) i (4 j ) T (j ) i = T (j ) i + T (j ) i T (j ) i (4 j ). Exmple 5. Compute Solution.8 sin x dx using Romberg s integrtion. x h.8 T = T () =.77.4 T =.76876,. T 4 =.776, T () 4 =.7795 T () 4 =.7796, T () 8 =.7795,. T 8 =.77887, T () 8 = Adptive Qudrture An dptive qudrture lgorithm uses one or two bsic qudrture rules, nmely, the trpezoidl rule nd Simpson s rule, nd determines the subintervl sizes so tht the computed result meets some prescribed ccurcy requirement. In this wy, n ttempt is mde to provide result with the prescribed ccurcy t the lowest cost possible. By cost we men computer time, which in turn is directly proportionl to the number of function evlutions necessry to obtin the result. The user of n dptive qudrture routine specifies the intervl [, b], provides subroutine which computes the function f(x) for ny x [, b], nd chooses tolernce ɛ. The dptive routine ttempts to compute quntity Q such tht Q f(x)dx ɛ.

12 Chp. 5. Numericl Integrtion CS44 Clss Notes 8 b Smll intervl size used in these regions Figure 5.7: Adptive qudrture. Exmple 5. Consider the problem of pproximtion I = prescribed tolernce ɛ. H f(x)dx using n dptive Simpson s rule with H/ H/4 H/8 Figure 5.8: Division of intervl [, b] for dptive qudrture. We first pproximte I using the bsic Simpson s rule over the intervl [, b], i.e., using one pnel, thus P = H [ ( f() + 4f + H ) ] + f(b). 6 Divide [, b] into two equl subintervls, nd pply Simpson s rule to ech. For the left hlf we get P = H [ ( f() + 4f + H ) ( + f + H )], 4 nd for the right hlf we obtin P = H Note tht we need only compute f ( + H 4 the previous level. Now, compute [ ( f + H ) + 4f ( + 34 ) ] H + f(b). ) nd f ( H), since the other vlues of f re vilble from Q = P + P b

13 Chp. 5. Numericl Integrtion CS44 Clss Notes 8 nd compre with P. If P Q ɛ for prescribed tolernce ɛ, report Q s the desired pproximtion to I. If not, set the right hlf [ + H, b] side for the moment nd proceed in the sme wy with the left hlf. In other words, compute P = H [ ( f() + 4f + H ) ( + f + H )], P = H [ ( f + H ) + 4f ( + 38 ) ( 4 4 H + f + H )]. We ccept (P + P ) s n pproximtion to +H/ f(x)df P (P + P ) ɛ, nd repet the process for the right hlf [ + H, b]. Thus, we hve constructed the following tree Level P : [, b] Level P : [, + H ] P : [ + H, b] Level 3 P : [, + H 4 ] P : [ + H 4, + H ] P : [ + H, + 3H 4 ] P : [ + 3H 4, b] If the bove test fils, however, set the right hlf of [, + H ] side nd repet the process for the left hlf [, + H 4 ]. In prctice, we hve to limit the number of levels used since the number of function evlutions increses rpidly. Hence, when we rech the mximum level permitted the lst pproximtion is ccepted nd we move to the right. In generl, let us ssume tht on n intervl [, + ] the bsic Simpson s rule yields n pproximtion S i to the true vlue I i = + f(x)dx. Then where < z i < + nd h i = +. Let S (L) i I i S i = (h i/) 5 f (iv) (z i ) (5.5) 9 nd S (R) i [, + hi ] nd [ + hi, +], respectively. Also, let Q i = S (L) i be the results of the bsic Simpson s rules on + S (R) i, then I i Q i = 4 (h i/) 5 f (iv) (y i ) (5.6) 9 where < y i < +. Assuming tht h i is smll enough so tht f (iv) x) is essentilly constnt, i.e., f (iv) (η) = f (iv) (y i ) = f (iv) (z i ), equtions (??) nd (??) yield or Q i S i = (h i /) 5 9 f (iv) (η) [ ] 4 I i = Qi + 4 [Q i S i ]. (5.7) Note tht (??) is one step of Richrdson s extrpoltion. In fct, similr to Appendix??, we cn show tht I i = S i + β h 4 i + β h 6 i ( ) ( ) 6 hi hi I i = Q i + β + β + nd I i = Q i + 4 [Q i S i ] + O(h 6 i ).

14 Chp. 5. Numericl Integrtion CS44 Clss Notes 83 Therefore, the bsic tsk of typicl routine is to bisect ech subintervl until the following inequlity is stisfied S i Q i 4 h i b ɛ where ɛ is the user specified tolernce. To see this, let Thus, Q Q = I = N i= N i= f(x)dx Q i, I i = = N i= xi+ f(x)dx. N (Q i I i ) i= N i= N i= Q i I i 4 Q i S i N ɛ h i = ɛ, b which is the desired gol! In this exmple we hve ssumed tht the routine is using n bsolute error criterion Q f(x)dx ɛ. It is usully preferble to use the reltive error tolernce Q b f(x)dx i= f(x) dx σ. Finlly, we would like to note tht it is not difficult to construct function f(x) for which given dptive qudrture routine fils. For exmple, the bove dptive Simpson s rule fils for the integrl I = b f(x)dx, f(x) = x (x ) (x ) (x 3) (x 4). In this cse, P =, nd both P = nd P = becuse f(x) hs roots t x =,,, 3, nd 4. And since P = P + P =, the qudrture rule will ssume the nswer to be ccurte! 5.6 Some Difficulties in Numericl Integrtion Discontinuous functions. When f(x) is hs discontinuity t the point c lying within the intervl [, b], we cn split the integrl s shown below f(x)dx = f(x)dx + c f(x)dx.

15 Chp. 5. Numericl Integrtion CS44 Clss Notes 84 f(x) c b Figure 5.9: Discontinuous function. Exmple 5.3 Compute the integrl I = f(x)dx for the pulse function { x /3 x < /3 f(x) = /3 - Figure 5.: Pulse function. Without splitting, the dptive Simpson s procedure converges slowly: P = /3, P = /3, P = /, P P + P, nd P = /4, P = /6, P P + P,... nd so on. On the other hnd, we get immedite convergence by splitting the integrl I = /3 f(x)dx + f(x)dx. /3

16 Chp. 5. Numericl Integrtion CS44 Clss Notes 85 Integrtion over infinite intervl. When the rnge of integrtion is infinite, i.e., I = we cn resort to chnge of vribles. Let y = x, then Exmple 5.4 Compute the infinite integrl Let y = x I = I = f(x)dx, >. / f e x ( ) dy y y. x dx. dy ; then x = y =, nd x = y =. Moreover, dx = y. Hence, I = e /y y dy = g(y)dy, where we cn esily evlute g(y) for ny y >. For y = we see tht e /y lim y y = lim y y( + y +! y + ) =. Singulrity. Singulrity t point c [, b] is tckled s follows: We split the function into two prts I = f(x)dx = f (x)dx + f (x)dx, where f (x) is smooth nd cn be integrted using n dptive procedure, nd f (x) contins the singulrity but cn be integrted nlyticlly. Exmple 5.5 Compute the integrl which hs singulrity t x =. Clerly, the function cn be split s follows I = cos x x dx cos x x = cos x x + x = f (x) + f (x). Even though f (x) hs singulrity t x =, its integrl cn be obtined nlyticlly, f (x)dx = x dx =. Integrl of f (x) cn now be hndled by n dptive qudrture routine provided we observe tht )] cos x lim = lim x [ + ( x x x x! + x4 4! x6 6! +... ] = lim [ x3/ + x7/ x/ + x! 4! 6! =, nd in the neighborhood of x =, cos x x.

17 Chp. 5. Numericl Integrtion CS44 Clss Notes 86 f(x) c b Figure 5.: Singulr function. 5.7 Appendix: Numericl Integrtion We wish to show tht the trunction error in trpezoidl rule is given by the following I{f} = f(x)dx = T (h) + α h + α h 4 + α 3 h 6 + where T (h) = h [ f + N i= f i + f N ], h = b N. f(x) x Figure 5.: Grph of f(x) = cos x x.

18 Chp. 5. Numericl Integrtion CS44 Clss Notes 87 Consider the pnel [, + ] with the midpoint y i = ( + + )/. Then Observing tht where h = +, then f(x) = f(y i ) + (x y i )f (y i ) + (x y i) xi+! (x y i ) m dx = f (y i ) + (x y i) 3 f (y i ) + 3! h m = m = h 3 m = m = 3 h 5 8 m = 4 However, Hence, xi+ f(x)dx = hf(y i ) + h3 4 f (y i ) + h5 9 f (iv) (y i ) + (5.8) f( ) = f(y i ) h f (y i ) + h 8 f (y i ) h3 48 f (y i ) + h4 384 f (iv) (y i )... f(+ ) = f(y i ) + h f (y i ) + h 8 f (y i ) + h3 48 f (y i ) + h4 384 f (iv) (y i ) + T i = h [f() + f(+ )] = hf(y i ) + h3 8 f (y i ) + h5 384 f (iv) (y i ) + Substituting in (??), we obtin Consequently, xi+ f(x)dx = f(x)dx = T i h3 f (y i ) h5 48 f (iv) (y i ) N in which < η, η < b. In other words, i= xi+ f(x)dx = T (h) h (b )f (η ) h4 48 (b )f (iv) (η ) I{f} = T (h) + α h + α h 4 +

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 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

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

3.4 Numerical integration

3.4 Numerical integration 3.4. Numericl integrtion 63 3.4 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 [,

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

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

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

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

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

Numerical Integration

Numerical Integration Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl

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

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

Discrete Least-squares Approximations

Discrete Least-squares Approximations Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

More information

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 + Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

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

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

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

Lecture 12: Numerical Quadrature

Lecture 12: Numerical Quadrature Lecture 12: Numericl Qudrture J.K. Ryn@tudelft.nl WI3097TU Delft Institute of Applied Mthemtics Delft University of Technology 5 December 2012 () Numericl Qudrture 5 December 2012 1 / 46 Outline 1 Review

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 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

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

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

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

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

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

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

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

MAT 772: Numerical Analysis. James V. Lambers

MAT 772: Numerical Analysis. James V. Lambers MAT 772: Numericl Anlysis Jmes V. Lmbers August 23, 2016 2 Contents 1 Solution of Equtions by Itertion 7 1.1 Nonliner Equtions....................... 7 1.1.1 Existence nd Uniqueness................ 7 1.1.2

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

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

Best Approximation in the 2-norm

Best Approximation in the 2-norm Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

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

Numerical integration. Quentin Louveaux (ULiège - Institut Montefiore) Numerical analysis / 10

Numerical integration. Quentin Louveaux (ULiège - Institut Montefiore) Numerical analysis / 10 Numericl integrtion Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis 2018 1 / 10 Numericl integrtion We consider definite integrls Z b f (x)dx better to it use if known! A We do not ssume tht

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

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

Calculus I-II Review Sheet

Calculus I-II Review Sheet Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

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

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

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

Chapter 2. Numerical Integration also called quadrature. 2.2 Trapezoidal Rule. 2.1 A basic principle Extending the Trapezoidal Rule DRAWINGS

Chapter 2. Numerical Integration also called quadrature. 2.2 Trapezoidal Rule. 2.1 A basic principle Extending the Trapezoidal Rule DRAWINGS S Cpter Numericl Integrtion lso clled qudrture Te gol of numericl integrtion is to pproximte numericlly. f(x)dx Tis is useful for difficult integrls like sin(x) ; sin(x ); x + x 4 Or worse still for multiple-dimensionl

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

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

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

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

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

x = b a N. (13-1) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is

x = b a N. (13-1) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is Jnury 28, 2002 13. The Integrl The concept of integrtion, nd the motivtion for developing this concept, were described in the previous chpter. Now we must define the integrl, crefully nd completely. According

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemann is the Mann! (But Lebesgue may besgue to differ.) Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

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

INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

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

Tangent Line and Tangent Plane Approximations of Definite Integral

Tangent Line and Tangent Plane Approximations of Definite Integral Rose-Hulmn Undergrdute Mthemtics Journl Volume 16 Issue 2 Article 8 Tngent Line nd Tngent Plne Approximtions of Definite Integrl Meghn Peer Sginw Vlley Stte University Follow this nd dditionl works t:

More information

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

Review of Riemann Integral

Review of Riemann Integral 1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.

More information

Trapezoidal Rule, n = 1, x 0 = a, x 1 = b, h = b a. f (x)dx = h 2 (f (x 0) + f (x 1 )) h3

Trapezoidal Rule, n = 1, x 0 = a, x 1 = b, h = b a. f (x)dx = h 2 (f (x 0) + f (x 1 )) h3 Trpezoidl Rule, n = 1, x 0 =, x 1 = b, h = b f (x)dx = h 2 (f (x 0) + f (x 1 )) h3 12 f (ξ). Simpson s Rule: n = 3, x 0 =, x 1 = +b 2, x 2 = b, h = b 2. Qudrture Rule, double node t x 1 P 3 (x)dx = f (x

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

Chapter 1. Basic Concepts

Chapter 1. Basic Concepts Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes (469-399)

More information

Lecture 1: Introduction to integration theory and bounded variation

Lecture 1: Introduction to integration theory and bounded variation Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You

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

31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes

31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be

More information

Chapters 4 & 5 Integrals & Applications

Chapters 4 & 5 Integrals & Applications Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

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. 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

Main topics for the First Midterm

Main topics for the First Midterm Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function? Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

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

Solutions to Assignment #8

Solutions to Assignment #8 Mth 1 Numericl Anlysis (Bueler) December 9, 29 Solutions to Assignment #8 Problems 64, exercise 14: The nswer turns out to be yes, which mens tht I hve to be orgnized in writing it up There re lot of fcts

More information

Taylor Polynomial Inequalities

Taylor Polynomial Inequalities Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil

More information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

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

6.5 Numerical Approximations of Definite Integrals

6.5 Numerical Approximations of Definite Integrals Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,

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

B.Sc. in Mathematics (Ordinary)

B.Sc. in Mathematics (Ordinary) R48/0 DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 B.Sc. in Mthemtics (Ordinry) SUPPLEMENTAL EXAMINATIONS 01 Numericl Methods Dr. D. Mckey Dr. C. Hills Dr. E.A. Cox Full mrks for complete nswers

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

Indefinite Integral. Chapter Integration - reverse of differentiation

Indefinite Integral. Chapter Integration - reverse of differentiation Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

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

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

Integration Techniques

Integration Techniques Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u

More information