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

Save this PDF as:

Size: px
Start display at page:

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

Transcription

1 Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October Definite Integrls In this section we revisit the definite integrl tht you were introduced to when you first studied clculus. You undoubtedly lerned tht given positive function f over n intervl [, b] the definite integrl f(x) dx provided it ws defined, ws number equl to the re under the grph of f over [, b]. You lso likely lerned tht the definite integrl ws defined s limit of Riemnn sums. The Riemnn sums you most likely used involved prtitioning [, b] into n uniform subintervls of length (b )/n nd evluting f t either the right-hnd endpoint, the left-hnd endpoint, or the midpoint of ech subintervl. At the time your understnding of the notion of limit ws likely more intuitive thn rigorous. In this section we present rigorous development of the definite integrl built upon the rigorous understnding of limit tht you hve studied erlier in this course Prtitions nd Drboux Sums. In the pproch tken here, we will consider very generl prtitions of the intervl [, b], not just those with uniform subintervls. Definition 1.1. Let [, b] R. A prtition of the intervl [, b] is specified by n N, nd {x i } n [, b] such tht = x 0 < x 1 < < x n 1 < x n = b. This prtition is denoted P = [x 0, x 1,, x n 1, x n ]. Ech x i for i = 0,, n is clled prtition point of P, nd for ech i = 1,, n the intervl [x i 1, x i ] is clled i th subintervl induced by P. The prtition thickness, denoted P, is defined by P = mx { x i x i 1 : i = 1,, n }. The pproch tken here is not bsed on Riemnn sums, but rther on Drboux sums. This is becuse Drboux sums re well-suited for nlysis by the tools we hve developed to estblish the existence of limits. We will be ble to recover results bout Riemnn sums becuse, s we will show, every Riemnn sum is bounded by two Drboux sums. 1

2 2 Let f : [, b] R be bounded. Set (1) m = inf { f(x) : x [, b] }, M = sup { f(x) : x [, b] }. Becuse f is bounded, one knows tht < m M <. Let P = [x 0,, x n ] be prtition of [, b]. For ech i = 1,, n set m i = inf { f(x) : x [x i 1, x i ] }, Clerly m m i M i M. M i = sup { f(x) : x [x i 1, x i ] }. Definition 1.2. The lower nd upper Drboux sums ssocited with the function f nd prtition P re respectively defined by (2) L(f, P) = U(f, P) = m i (x i x i 1 ), M i (x i x i 1 ). Clerly, the Drboux sums stisfy the bounds (3) m (b ) L(f, P) U(f, P) M (b ). These inequlities will ll be equlities when f is constnt. Remrk. A Riemnn sum ssocited with the prtition P is specified by selecting qudrture point q i [x i 1, x i ] for ech i = 1,, n. Let Q = (q 1,, q n ) be the n-tuple of qudrture points. The ssocited Riemnn sum is then (4) R(f, P, Q) = f(q i ) (x i x i 1 ). It is esy to see tht for ny choice of qudrture points Q one hs the bounds (5) L(f, P) R(f, P, Q) U(f, P). Moreover, one cn show tht L(f, P) = inf { R(f, P, Q) : Q re qudrture points for P }, U(f, P) = sup { R(f, P, Q) : Q re qudrture points for P }. The bounds (5) re thereby shrp.

3 1.2. Refinements. We now introduce the notion of refinement of prtition. Definition 1.3. Given prtition P of n intervl [, b], prtition P of [, b] is clled refinement of P provided every prtition point of P is prtition point of P. If P = [x 0, x 1,, x n 1, x n ] nd P is refinement of P then P induces prtition of ech [x i 1, x i ], which we denote by Pi. For exmple, if P = [x 0, x 1,, x n 1, x n ] with x j i = x i for ech i = 0,, n then Pi = [x j i 1,, x j i ]. Observe tht (6) L(f, P ) = L(f, Pi ), U(f, P ) = U(f, Pi ). Moreover, upon pplying the bounds (3) to P i for ech i = 1,, n, we obtin the bounds (7) m i (x i x i 1 ) L(f, P i ) U(f, P i ) M i (x i x i 1 ). This observtion is key to the proof of the following. Lemm 1.1. Refinement Lemm. Let f : [, b] R be bounded. Let P be prtition of [, b] nd P be refinement of P. Then (8) L(f, P) L(f, P ) U(f, P ) U(f, P). 3 Proof. It follows from (2), (7), nd (6) tht L(f, P) = m i (x i x i 1 ) L(f, P i ) = L(f, P ) U(f, P ) = U(f, P i ) M i (x i x i 1 ) = U(f, P) Comprisons. A key step in our development will be to develop comprisons of L(f, P 1 ) nd U(f, P 2 ) for ny two prtitions P 1 nd P 2, of [, b].

4 4 Definition 1.4. Given two prtitions, P 1 nd P 2, of [, b] define P 1 P 2 to be the prtition whose set of prtition points is the union of the prtition points of P 1 nd the prtition points of P 2. We cll P 1 P 2 the supremum of P 1 nd P 2. It is esy to rgue tht P 1 P 2 is the smllest prtition of [, b] tht is refinement of both P 1 nd P 2. Lemm 1.2. Comprison Lemm. Let f : [, b] R be bounded. Let P 1 nd P 2 be prtitions of [, b]. Then (9) L(f, P 1 ) U(f, P 2 ). Proof. Becuse P 1 P 2 is refinement of both P 1 nd P 2, it follows from the Refinement Lemm tht L(f, P 1 ) L(f, P 1 P 2 ) U(f, P 1 P 2 ) U(f, P 2 ). Becuse the prtitions P 1 nd P 2 on either side of inequlity (9) re independent, we my obtin shrper bounds by tking the supremum over P 1 on the right-hnd side, or the infimum over P 2 on the left-hnd side. Indeed, we prove the following. Lemm 1.3. Shrp Comprison Lemm. Let f : [, b] R be bounded. Let (10) L(f) = sup { L(f, P) : P is prtition of [, b] }, U(f) = inf { U(f, P) : P is prtition of [, b] }. Let P 1 nd P 2 be prtitions of [, b]. Then (11) L(f, P 1 ) L(f) U(f) U(f, P 2 ). Remrk. Becuse it is cler from (10) tht L(f) nd U(f) depend on [, b], strictly speking these quntities should be denoted L(f, [, b]) nd U(f, [, b]). This would be necessry if more thn one intervl ws involved in the discussion. However, tht is not the cse here. We therefore embrce the less cluttered nottion. Proof. If we tke the infimum of the right-hnd side of (9) over P 2, we obtin L(f, P 1 ) U(f). If we then tke the supremum of the left-hnd side bove over P 1, we obtin L(f) U(f). The bound (11) then follows.

5 1.4. Definition of the Definite Integrl. We re now redy to define the definite integrl. You will find different definitions of the definite integrl in different books. Here we will used the definition found in Fitzptrick s book. We will then give theorem tht shows this definition is equivlent to nother one commonly found in other books. Definition 1.5. Let f : [, b] R be bounded. Then f is sid to be integrble over [, b] whenever there exists unique A R such tht (12) L(f, P) A U(f, P) for every prtition P of [, b]. In this cse we cll A the definite integrl of f over [, b] nd denote it by f. Remrk. The Shrp Comprison Lemm shows tht (12) holds for every A [L(f), U(f)]. The key thing to be estblished when using the bove definition is therefore the uniqueness of such n A. We now give the following chrcteriztions of integrbility. Theorem 1.1. Integrbility Theorem. Let f : [, b] R be bounded. Then the following re equivlent: (1) f is integrble over [, b]; (2) L(f) = U(f); (3) for every ǫ > 0 there exists prtition P of [, b] such tht 0 U(f, P) L(f, P) < ǫ. Proof. We first show tht (1) = (2). Suppose tht (2) is flse. Then L(f) < U(f). Observe tht the Shrp Comprison Lemm shows tht for every A [L(f), U(f)] one hs L(f, P 1 ) A U(f, P 2 ) for ny prtitions P 1 nd P 2 of [, b]. Hence, there re mny vlues of A tht stisfy (12), whereby f is not integrble. It follows tht (1) = (2). Next we show tht (2) = (3). Let ǫ > 0. By the definition (10) of L(f) nd U(f), we cn find prtitions P 1 nd P 2 of [, b] such tht L(f) ǫ 2 < L(f, P 1 ) L(f), U(f) U(f, P 2 ) < U(f) + ǫ 2. Now let P = P 1 P 2. Becuse the Comprison Lemm implies tht L(f, P 1 ) L(f, P) nd U(f, P) U(f, P 2 ), it follows from the bove 5

6 6 inequlities tht L(f) ǫ 2 < L(f, P) L(f), U(f) U(f, P) < U(f) + ǫ 2. Hence, if L(f) = U(f) one thereby concludes tht ( 0 U(f, P) L(f, P) < U(f) + ǫ ) ( L(f) ǫ ) = ǫ. 2 2 This shows tht (2) = (3). Finlly, we show tht (3) = (1). Suppose tht (1) is flse. Then by the Shrp Comprison Lemm there exists A 1 nd A 2 such tht L(f, P) A 1 < A 2 U(f, P) for every prtition P of [, b]. One thereby hs tht U(f, P) L(f, P) A 2 A 1 for every prtition P of [, b]. Hence, (3) must be flse. It follows tht (1) = (2). Remrk. Property (3) of the Integrbility Theorem provides very useful criterion for estblishing the integrbility of function f. We will exploit this in the next section. Remrk. It follows from the Integrbility Theorem tht one could well dopt the following lterntive definition of the definite integrl. Definition 1.5. Let f : [, b] R be bounded. Then f is sid to be integrble over [, b] whenever L(f) = U(f). In this cse we cll this common vlue the definite integrl of f over [, b] nd denote it by f. This is definition of the definite integrl tht is commonly found in textbooks. 2. Integrble Functions nd Integrls In this section we use the Integrbility Theorem to develop criteri to identify integrble functions. We lso begin the tsk of evluting definite integrls Evluting Integrls vi Riemnn Sums. We now mke the connection with the notion of definite integrl s the limit of sequence of Riemnn sums.

7 Theorem 2.1. Riemnn Sums Convergence Theorem. Let f : [, b] R be bounded. Let {P n } n=1 be sequence of prtitions of [, b] such tht ( (13) lim U(f, P n ) L(f, P n ) ) = 0. n Let Q n be ny qudrture set ssocited with P n. Then f is integrble over [, b] nd (14) f = lim n R(f, P n, Q n ), where the Riemnn sums R(f, P, Q) re defined by (4). Remrk. The content of this theorem is tht once one hs found sequence of prtitions P n such tht (13) holds, then the integrl f exists nd my be evluted s the limit of ny ssocited sequence of Riemnn sums (14). This theorem thereby splits the tsk of evluting definite integrls into two steps. The first step is by fr the esier. It is rre integrnd f for which one cn find sequence of Riemnn tht llow one to evlute the limit in (14). Proof. Given (13), the fct tht f is integrble over [, b] follows directly from criterion (3) of the Integrbility Theorem. The bounds on Riemnn sums given by (5) yield the inequlities L(f, P n ) R(f, P n, Q n ) U(f, P n ), while, becuse f is integrble, we lso hve the inequlities L(f, P n ) It follows from these inequlities tht L(f, P n ) U(f, P n ) L(f, P n ) which implies tht R(f, P n, Q n ) f U(f, P n ). R(f, P n, Q n ) U(f, P n ) f f f U(f, P n ) L(f, P n ), f U(f, P n ) L(f, P n ). Becuse (13) sttes tht the right-hnd side bove vnishes s n tends to, the limit (14) follows. 7

8 Monotone nd Piecewise Monotone Functions. We now use the Riemnn Sums Convergence Theorem to show tht the clss of integrble functions includes the clss of monotone functions. Recll tht this clss is defined s follows. Definition 2.1. A function f : [, b] R is sid to be monotoniclly incresing provided tht x < y = f(x) f(y) for every x, y [, b]. A function f : [, b] R is sid to be monotoniclly decresing provided tht x < y = f(x) f(y) for every x, y [, b]. If function is either monotoniclly incresing or monotoniclly decresing then it is sid to be monotone. It is clssicl fct tht monotone function over [, b] is continuous t ll but t most countble number of points where it hs jump discontinuity. Theorem 2.2. Let f : [, b] R be monotone. Then f is integrble over [, b]. Moreover, for ny sequences P n of prtitions of [, b] nd Q n of ssocited qudrture points such tht P n 0 s n, one hs tht f = lim R(f, P n, Q n ), n where the Riemnn sums R(f, P, Q) re defined by (4). Proof. For ny prtition P = [x 0,, x n ] we hve the following bsic estimte. Becuse f is monotone, over ech subintervl [x i 1, x i ] one hs tht M i m i = f(x i ) f(x i 1 ). We thereby obtin the bsic estimte 0 U(f, P) L(f, P) = (M i m i ) (x i x i 1 ) P (M i m i ) = P = P f(b) f(), f(xi ) f(x i 1 ) where P = mx{x i x i 1 : i = 1,, n} is the thickness of P. Here we hve used the fct tht, becuse f is monotone, the terms f(x i ) f(x i 1 ) re either ll nonnegtive, or ll nonpositive. This

9 fct llows us to pss the bsolute vlue outside the sum, which then telescopes. We now pply the bove bsic estimte to our sequence P n of prtitions, which shows tht 0 U(f, P n ) L(f, P n ) P n f(b) f() 0 s n. The result then follows from the Riemnn Sums Convergence Theorem. Exmple. One cn use this theorem to show tht for every k N the function x x k is integrble over [0, b] nd tht [ b (15) x k k+1 ] dx = lim i k = bk+1 n n k+1 k The detils of this clcultion re presented in the book for the cses k = 0, 1, 2 with b = 1. Here we present the generl cse. Define S k (n) = i k. In order to prove (15) we must estblish the limit 1 (16) lim n n k+1 Sk (n) = 1 k + 1. We do this below by induction on k. Proof. Clerly S 0 (n) = n, so tht limit (16) holds for k = 0. Now ssume tht for some l 1 limit (16) holds for every k < l. By telescoping sum, binomil expnsion, nd the definition of S k (n), one obtins the identity (n + 1) l+1 1 = = = [ (i + 1) l+1 i l+1] l j=0 l j=0 (l + 1)! j!(l j + 1)! ij (l + 1)! j!(l j + 1)! Sj (n) = (l + 1) S l (n) + l 1 j=0 (l + 1)! j!(l j + 1)! Sj (n). 9

10 10 Upon solving for S l (n) nd dividing by n l+1, we obtin the reltion (17) [ 1 n l+1 Sl (n) = 1 (n + 1) l+1 1 l + 1 n l+1 n l+1 Becuse we know l 1 j=0 (l + 1)! j!(l j + 1)! (n + 1) l+1 1 lim = 1, lim n n l+1 n n = 0, l+1 nd becuse, by the induction hypothesis, we know lim n 1 n l+1 Sj (n) = 0 for every j < l, ] 1 n l+1 Sj (n). we cn pss to the n limit in reltion (17). We thereby estblish tht limit (16) holds for k = l. We now use the Riemnn Sums Convergence Theorem to show tht the clss of integrble functions includes the clss of piecewise monotone functions. Recll tht this clss is defined s follows. Definition 2.2. A function f : [, b] R is sid to be piecewise monotone over [, b] provided there exists prtition [p 0,, p m ] of [, b] such tht f is monotone over [p j 1, p j ] for every j = 1,, m. Theorem 2.3. Piecewise Monotone Integrbility Theorem. Let f : [, b] R be piecewise monotone over [, b]. Then f is integrble over [, b]. Moreover, for ny sequences P n of prtitions of [, b] nd Q n of ssocited qudrture points such tht P n 0 s n, one hs tht f = lim n R(f, P n, Q n ), where the Riemnn sums R(f, P, Q) re defined by (4). Proof. Let [p 0,, p m ] be prtition of [, b] such tht f is monotone over [p j 1, p j ] for ech j = 1,, m. Let P = [x 0,, x n ] be ny prtition of [, b]. The key step will be to estblish the bound (18) (M i m i ) m f(pj ) f(p j 1 ). j=1

11 11 Once this is done we cn obtin the bsic estimte 0 U(f, P) L(f, P) = P P (M i m i 1 ) (M i m i ) (x i x i 1 ) m f(pj ) f(p j 1 ), where P = mx{x i x i 1 : i = 1,, n} is the thickness of P. By then pplying the bove bsic estimte to our sequence P n of prtitions, we see tht 0 U(f, P n ) L(f, P n ) m P n f(pj ) f(p j 1 ) 0 s n. The result would then follows from the Riemnn Sums Convergence Theorem. All tht remins to be done is estblish the bound (18). This is esy to do when P is refinement of [p 0,, p m ]. When P is not refinement of [p 0,, p m ] one simply replces P with P [p 0,, p m ]. We leve the detils of these rguments s n exercise Piecewise Integrbility. A key tool for building up the clss of integrble functions is the the following lemm. Lemm 2.1. Piecewise Integrbility Lemm. Let f : [, b] R be bounded. Let P = [p 0,, p k ] be prtition of [, b] such tht f is integrble over [p i 1, p i ] for every i = 1,, k. Then f is integrble over [, b]. Moreover, (19) f = k pi p i 1 f. Proof. Let ǫ > 0. Becuse f is integrble over [p i 1, p i ] there exists prtition P i of [p i 1, p i ] such tht 0 U(f, P i ) L(f, P i ) < ǫ k.

12 12 Let P be the refinement of P such tht P i is the induced prtition of [p i 1, p i ]. One then sees tht 0 U(f, P ) L(f, P ) k ( = U(f, P i ) L(f, Pi )) < k ǫ k = ǫ. Hence, by item (3) of the Integrbility Theorem, f is integrble. Let P be prtition of [, b]. Without loss of generlity we my ssume tht P is refinement of P, otherwise pss to the prtition P P. Let Pi be the induced prtition of [p i 1, p i ]. Becuse f is integrble over [p i 1, p i ] we hve tht pi L(f, Pi ) f U(f, Pi ). p i 1 Summing these inequlities over i = 1,, k yields k pi L(f, P ) f U(f, P ). p i 1 Formul (19) then follows Piecewise Continuous Functions. We now show tht ll functions tht re piecewise continuous over [, b] re lso intergrble over [, b]. We first recll the definition of piecewise continuous function. Definition 2.3. A function f : [, b] R is sid to be piecewise continuous if it is bounded nd there exists prtition P = [x 0,, x n ] of [, b] such tht f is continuous over (x i 1, x i ) for every i = 1,, n. We remrk tht piecewise continuous functions re discontinuous t only finite number of points. Still, the clss of piecewise continuous functions includes some firly wild functions. For exmple, it contins the function 1 + sin(1/x) if x > 0, f(x) = 4 if x = 0, 1 + sin(1/x) if x < 0, considered over [ 1, 1]. As wild s this function looks, it is continuous everywhere except t the point x = 0. We will need two lemms. Lemm 2.2. Let f : [, b] R be continuous. Then f is integrble over [, b].

13 Proof. Let ǫ > 0. Becuse f is uniformly continuous over [, b], there exists δ > 0 such tht x y < δ = f(x) f(y) < ǫ for every x, y [, b]. b Pick n N such tht b n < δ. Let P = [x 0,, x n ] be the prtition of [, b] with x i = + i b for every i = 0,, n. n For every i = 1,, n over the subintervl [x i 1, x i ] one hs x i x i 1 = b n, M i m i ǫ b. One thereby sees tht 0 U(f, P) L(f, P) = (M i m i )(x i x i 1 ) = b n < b n (M i m i ) ǫ b = b n n ǫ b = ǫ. Hence, by item (3) of the Integrbility Theorem, f is integrble. Lemm 2.3. Let f : [, b] R be bounded nd f : (, b) R be continuous. Then f is integrble over [, b]. Proof. Let ǫ > 0. Let δ > 0 such tht ǫ δ < 3(M m), δ < b. 2 where, s before, m nd M re defined by (1). Becuse f is continuous over [ + δ, b δ], by the previous lemm it is integrble over [ + δ, b δ]. By the Integrbility Theorem, there exists prtition P δ of [ + δ, b δ] such tht 0 U(f, P δ ) L(f, P δ ) < ǫ 3. Let P be the prtition of [, b] given by P = [x 0, x 1,, x n 1, x n ] where P δ = [x 1,, x n 1 ]. One hs x 1 x 0 = x n x n 1 = δ, M 1 m 1 < M m, M n m n < M m. 13

14 14 One thereby sees tht 0 U(f, P) L(f, P) = (M i m i )(x i x i 1 ) n 1 = (M 1 m 1 )δ + (M n m n )δ + (M i m i )(x i x i 1 ) i=2 = (M 1 m 1 )δ + (M n m n )δ + U(f, P δ ) L(f, P δ ) ǫ < 2 (M m) 3(M m) + ǫ 3 = ǫ. Hence, by item (3) of the Integrbility Theorem, f is integrble. Theorem 2.4. Let f : [, b] R be piecewise continuous. Then f is integrble over [, b]. Proof. The theorem follows from the previous lemm nd the Piecewise Integrbility Lemm. 3. The First Fundmentl Theorem of Clculus The business of evluting integrls by tking limits of Riemnn sums is usully either difficult or impossible. However, s you hve known since you first studied integrtion, for mny integrnds there is must esier wy. Theorem 3.1. First Fundmentl Theorem of Clculus. Let f : [, b] R be integrble. Suppose tht F : [, b] R is continuous, tht F : (, b) R is differentible, nd tht (20) F (x) = f(x) for every x (, b). Then f = F(b) F(). Remrk. This theorem essentilly reduces the problem of evluting definite integrls to tht of finding n explicit solution of the differentil eqution (20). While such n explicit solution cnnot lwys be found, for wide clss of integrnds f. Proof. We must show tht for every prtition P of [, b] one hs (21) L(f, P) F(b) F() U(f, P). Let P = [x 0,, x n ] be n rbitrry prtition of [, b]. For every i = 1,, n one knows tht F : [x i 1, x i ] R is continuous, nd tht

15 F : (x i 1, x i ) R is differentible. Then by the Lgrnge Men Vlue Theorem there exists q i (x i 1, x i ) such tht F(x i ) F(x i 1 ) = F (q i ) (x i x i 1 ) = f(q i ) (x i x i 1 ). Becuse m i f(q i ) M i, we see from the bove tht m i (x i x i 1 ) F(x i ) F(x i 1 ) M i (x i x i 1 ). Finlly, dding these inequlities yields (21). The following is n immedite corollry of the First Fundmentl Theorem of Clculus. Corollry 3.1. Let F : [, b] R be continuous, F : (, b) R be differentible, nd F : (, b) R be continuous nd bounded. Let f be ny extension of F to [, b]. Then f = F(b) F(). Exmple. Let F be defined over [0, 1] by { x cos(log(1/x)) if 0 < x 1, F(x) = 0 if x = 0. Then F is continuous over [0, 1] nd differentible over (0, 1] with F (x) = cos(log(1/x)) + sin(log(1/x)). As this function is bounded, we hve 1 0 [cos(log(1/x)) + sin(log(1/x))] dx = F(1) F(0) = 0. Here the integrnd cn be ssigned ny vlue t x = 0. 15

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

Properties of the Riemann Integral

Properties of the Riemann Integral Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2

More information

Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015

Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015 Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

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

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

7.2 Riemann Integrable Functions

7.2 Riemann Integrable Functions 7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous

More information

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)

Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar) Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of

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

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

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q. Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

More information

MAA 4212 Improper Integrals

MAA 4212 Improper Integrals Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

More information

Riemann Stieltjes Integration - Definition and Existence of Integral

Riemann Stieltjes Integration - Definition and Existence of Integral - Definition nd Existence of Integrl Dr. Adity Kushik Directorte of Distnce Eduction Kurukshetr University, Kurukshetr Hryn 136119 Indi. Prtition Riemnn Stieltjes Sums Refinement Definition Given closed

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

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

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

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

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

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

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

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

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

Presentation Problems 5

Presentation Problems 5 Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).

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

0.1 Properties of regulated functions and their Integrals.

0.1 Properties of regulated functions and their Integrals. MA244 Anlysis III Solutions. Sheet 2. NB. THESE ARE SKELETON SOLUTIONS, USE WISELY!. Properties of regulted functions nd their Integrls.. (Q.) Pick ny ɛ >. As f, g re regulted, there exist φ, ψ S[, b]:

More information

7.2 The Definition of the Riemann Integral. Outline

7.2 The Definition of the Riemann Integral. Outline 7.2 The Definition of the Riemnn Integrl Tom Lewis Fll Semester 2014 Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Outline Upper nd

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

arxiv:math/ v2 [math.ho] 16 Dec 2003

arxiv:math/ v2 [math.ho] 16 Dec 2003 rxiv:mth/0312293v2 [mth.ho] 16 Dec 2003 Clssicl Lebesgue Integrtion Theorems for the Riemnn Integrl Josh Isrlowitz 244 Ridge Rd. Rutherford, NJ 07070 jbi2@njit.edu Februry 1, 2008 Abstrct In this pper,

More information

The Henstock-Kurzweil integral

The Henstock-Kurzweil integral fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft

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

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

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

Calculus in R. Chapter Di erentiation

Calculus in R. Chapter Di erentiation Chpter 3 Clculus in R 3.1 Di erentition Definition 3.1. Suppose U R is open. A function f : U! R is di erentible t x 2 U if there exists number m such tht lim y!0 pple f(x + y) f(x) my y =0. If f is di

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

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

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition

More information

Principles of Real Analysis I Fall VI. Riemann Integration

Principles of Real Analysis I Fall VI. Riemann Integration 21-355 Principles of Rel Anlysis I Fll 2004 A. Definitions VI. Riemnn Integrtion Let, b R with < b be given. By prtition of [, b] we men finite set P [, b] with, b P. The set of ll prtitions of [, b] will

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

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

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

For a continuous function f : [a; b]! R we wish to define the Riemann integral

For a continuous function f : [a; b]! R we wish to define the Riemann integral Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This

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

11 An introduction to Riemann Integration

11 An introduction to Riemann Integration 11 An introduction to Riemnn Integrtion The PROOFS of the stndrd lemms nd theorems concerning the Riemnn Integrl re NEB, nd you will not be sked to reproduce proofs of these in full in the exmintion in

More information

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a). The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

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

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

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

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

1. On some properties of definite integrals. We prove

1. On some properties of definite integrals. We prove This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.

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

1.3 The Lemma of DuBois-Reymond

1.3 The Lemma of DuBois-Reymond 28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr

More information

ON THE C-INTEGRAL BENEDETTO BONGIORNO

ON THE C-INTEGRAL BENEDETTO BONGIORNO ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives

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

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

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals. MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

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

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

Advanced Calculus I (Math 4209) Martin Bohner

Advanced Calculus I (Math 4209) Martin Bohner Advnced Clculus I (Mth 4209) Spring 2018 Lecture Notes Mrtin Bohner Version from My 4, 2018 Author ddress: Deprtment of Mthemtics nd Sttistics, Missouri University of Science nd Technology, Roll, Missouri

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

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls

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

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

arxiv: v1 [math.ca] 11 Jul 2011

arxiv: v1 [math.ca] 11 Jul 2011 rxiv:1107.1996v1 [mth.ca] 11 Jul 2011 Existence nd computtion of Riemnn Stieltjes integrls through Riemnn integrls July, 2011 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde

More information

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer. Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points

More information

FUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 (

FUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 ( FUNDAMNTALS OF RAL ANALYSIS by Doğn Çömez III. MASURABL FUNCTIONS AND LBSGU INTGRAL III.. Mesurble functions Hving the Lebesgue mesure define, in this chpter, we will identify the collection of functions

More information

II. Integration and Cauchy s Theorem

II. Integration and Cauchy s Theorem MTH6111 Complex Anlysis 2009-10 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.

More information

ODE: Existence and Uniqueness of a Solution

ODE: Existence and Uniqueness of a Solution Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

Week 7 Riemann Stieltjes Integration: Lectures 19-21

Week 7 Riemann Stieltjes Integration: Lectures 19-21 Week 7 Riemnn Stieltjes Integrtion: Lectures 19-21 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0

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

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

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

Math 61CM - Solutions to homework 9

Math 61CM - Solutions to homework 9 Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ

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

IMPORTANT THEOREMS CHEAT SHEET

IMPORTANT THEOREMS CHEAT SHEET IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.

More information

Chapter 6. Riemann Integral

Chapter 6. Riemann Integral Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

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

F (x) dx = F (x)+c = u + C = du,

F (x) dx = F (x)+c = u + C = du, 35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil

More information

LECTURE. INTEGRATION AND ANTIDERIVATIVE.

LECTURE. INTEGRATION AND ANTIDERIVATIVE. ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

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

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

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral.

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral. MATH 409 Advnced Clculus I Lecture 18: Drboux sums. The Riemnn integrl. Prtitions of n intervl Definition. A prtition of closed bounded intervl [, b] is finite subset P [,b] tht includes the endpoints

More information

Week 10: Riemann integral and its properties

Week 10: Riemann integral and its properties Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the

More information

Preliminaries From Calculus

Preliminaries From Calculus Chpter 1 Preliminries From Clculus Stochstic clculus dels with functions of time t, t T. In this chpter some concepts of the infinitesiml clculus used in the sequel re given. 1.1 Functions in Clculus Continuous

More information

Appendix to Notes 8 (a)

Appendix to Notes 8 (a) Appendix to Notes 8 () 13 Comprison of the Riemnn nd Lebesgue integrls. Recll Let f : [, b] R be bounded. Let D be prtition of [, b] such tht Let D = { = x 0 < x 1

More information

Riemann Integrals and the Fundamental Theorem of Calculus

Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

More information

Chapter 28. Fourier Series An Eigenvalue Problem.

Chapter 28. Fourier Series An Eigenvalue Problem. Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why

More information

Math 324 Course Notes: Brief description

Math 324 Course Notes: Brief description Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd

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

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

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

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

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

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued

More information

CHAPTER 4 MULTIPLE INTEGRALS

CHAPTER 4 MULTIPLE INTEGRALS CHAPTE 4 MULTIPLE INTEGAL The objects of this chpter re five-fold. They re: (1 Discuss when sclr-vlued functions f cn be integrted over closed rectngulr boxes in n ; simply put, f is integrble over iff

More information

Math Advanced Calculus II

Math Advanced Calculus II Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused

More information

arxiv: v1 [math.ca] 7 Mar 2012

arxiv: v1 [math.ca] 7 Mar 2012 rxiv:1203.1462v1 [mth.ca] 7 Mr 2012 A simple proof of the Fundmentl Theorem of Clculus for the Lebesgue integrl Mrch, 2012 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde

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

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

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