Controllable Microfluidic Production of Multicomponent Multiple Emulsions

Size: px
Start display at page:

Download "Controllable Microfluidic Production of Multicomponent Multiple Emulsions"

Transcription

1 Supplementry Mteril (ESI) or L on Chip This journl is The Royl Society o Chemistry 0 Controllle Microluiic Prouction o Multicomponent Multiple Emulsions Supplementry Mteril Wei Wng, Rui Xie *, Xio-Jie Ju, To Luo, Li Liu, vi A. Weitz, Ling-Yin Chu * School o Chemicl Engineering, Sichun University, Chengu, Sichun, Chin Tel: ; E-mil: chuly@scu.eu.cn; xierui@scu.eu.cn School o Engineering n Applie Sciences/eprtment o Physics, Hrvr University, Cmrige, Msschusetts, USA. Prt S. Experimentl S.. Friction o microluiic evice The microluiic evice is se on three sic uiling locks, s shown in Fig.. Cylinricl cpillries with outer imeter o.0 mm (AIT glss) re use s the injection tue n collection tue. The en o the injection tue is tpere y micropuller (Nrishige, Jpn) n then juste y microorge (Nrishige, Jpn). Squre cpillry tues with inner imension o.0 mm (Vitrocom, USA) re use to ensure the coxility o the cylinricl cpillries neste within them. For the rop mker, the injection tue n collection tue re ligne insie the squre cpillry tue, with the tpere en o the injection tue inserte into the collection tue. The coxil co-low geometry o rop mker is ensure y mtching the outer imeters o the cylinricl cpillries to the inner imensions o the squre cpillry tues, s shown in Fig.S. The isperse lui A is pumpe into the injection tue, n the continuous lui B is pumpe through the outer coxil region etween the squre cpillry tue n the cylinricl injection tue. The coxil low o these two luis results in the ormtion o monoisperse roplets in the collection tue. For the connector, two cylinricl cpillries re integrte to uil Y-shpe connector. For the liqui extrctor, the injection tue n collection tue re ligne insie the squre cpillry tue ut with the tpere en o the injection tue out o the collection tue. Friction o the microluiic evices or vrious multicomponent multiple emulsions is chieve y selective oning o the rop mker, connector n liqui extrctor. For exmple, to ricte the microluiic evice in Fig., uiling locks re one together y using PVC tue to connect the cylinricl cpillries o every two uiling locks, s shown in Fig.S. S

2 Fig.S Schemtic igrm o the rop mker or generting monoisperse roplets. Illustrtions -, -, c-c n - re cross-section imges o the rop mker in relevnt positions, which clerly show how the cylinricl cpillries n squre cpillry tues re ssemle in the rop mker. Flui A is the isperse phse, n lui B is the continuous phse. Fig.S Cpillry ssemly o the uiling locks or the riction o microluiic evice or quruple-component oule emulsions. A PVC tue is use to connect the cylinricl cpillries o every two uiling locks. S.. Genertion o multicomponent multiple emulsions We generte multicomponent multiple emulsions y operting luis o queous phse n oil phse into n out o the microluiic evice through syringe pumps. With 5 % (w/v) polyglycerol polyricinolete (PGPR90, nisco) s surctnt, soyen oil n enzyl enzote re use s two kins o oil phses. Aqueous solution contining % (w/v) Pluronic F7 (Sigm-Alrich) n 5 % (w/v) glycerol is use s queous phse. Either 0. S

3 % (w/v) Sun re or 0. % (w/v) Sun lck is use to color the oil phse, n 0.5 % (w/v) methylene lue is use to color the queous phse. For quruple-component oule emulsions, inner luis F - n F - re soyen oil phse with n without Sun re respectively, F -, F -, F -C n F -E re queous phses, n F -M is soyen oil phse, s shown in Fig.. The vrition rnges or the low rtes re - 50~00 μl h -, - 0~00 μl h -, - 00~00 μl h -, - 00~700 μl h -, -C 50~00 μl h -, -E 00~700 μl h - n -M 4000~000 μl h -. For quintuple-component oule emulsions, F -, F - n F - re enzyl enzote oil phse lele with Sun re, Sun lck n without ny ye respectively, F - is queous phse, n the other luis re the sme s those o quruple-component oule emulsions, s shown in Fig.4. The vrition rnges or the low rtes re - 00~00 μl h -, - 00~00 μl h -, - 00~00 μl h -, - 500~00 μl h -, - 500~00 μl h -, - 500~00 μl h -, -C 00~500 μl h -, -E 00~700 μl h -, n -M 4000~000 μl h -. For quintuple-component triple emulsions, F - n F - re enzyl enzote oil phse with n without Sun re respectively, F -, F -, F -C n F -E re queous phses with methylene lue, F -M is enzyl enzote oil phse, n F 4-M is queous phse, s shown in Fig.4c. The vrition rnges or the low rtes re - 00~50 μl h -, - 00~00 μl h -, - 850~900 μl h -, - 700~800 μl h -, -C 00 μl h -, -E 0~400 μl h -, -M 00~4000 μl h - n 4-M 9000~000 μl h -. For sextuple-component triple emulsions, F - n F - re enzyl enzote oil phse with n without Sun re respectively, F - n F - re queous phse n queous phse with methylene lue respectively, F -, F -, F -C n F -E re enzyl enzote oil phses, n F 4-M is queous phse, s shown in Fig.5. The vrition rnges or the low rtes re - 0~400 μl h -, - 0~00 μl h -, - 400~000 μl h -, - 50~000 μl h -, - 700~00 μl h -, - 700~00 μl h -, -C 00~500 μl h -, -E 00~000 μl h - n 4-M 4000~0000 μl h -. Prt S. Results n iscussion S.. Flow-rte-epenent control over numer n rtio o ierent roplets in quruple-component oule emulsions For quruple-component oule emulsions, the inner re n trnsprent roplets re seprtely orme rom the rop mkers in ierent rnch chnnels, so the ormtion rte o ech kin o roplets cn e iniviully tune y chnging the relevnt low rtes, which S

4 results in regulr rry o re n trnsprent roplets with controlle rtio in the min chnnel, s shown in Fig.S. When the ormtion rtes o two kins o roplets re equl, the re n trnsprent roplets rrnge in the min chnnel one y one, where every one re roplet n one trnsprent roplet mkes perioic unit o the regulr roplet rry (Fig.S()). Brek-up in the intervls o perioic units les to co-encpsultion o two kins o roplets with controlle rtio t :. By chnging the low rte o the outer lui, every one, two n three perioic units cn e encpsulte in oule emulsions, resulte in quruple-component oule emulsions contining two, our n six roplets respectively, with the rtio (R ) o the re roplets to trnsprent roplets ixe t : (Fig.S()). By incresing the ormtion rte o the re roplets, regulr rrys with more re roplets in every perioic unit cn e otine (Fig.S()-()). Brek-up in the intervls o these perioic units results in quruple-component oule emulsions with R controlle t :, : n :, s shown in Fig.S(), (c) n (), respectively. Similrly, increse in the ormtion rte o the trnsprent roplets les to regulr rrys with more trnsprent roplets in every perioic unit (Fig.S(e)-(g)), n perioic rek-up o these rrys results in quruple-component oule emulsions with R controlle t :, : n :, s shown in Fig.S(e), () n (g), respectively. Thereore, the numer n rtio o the encpsulte roplets with ierent contents cn e inepenently n precisely controlle y justing the low rtes o relevnt luis. Fig.S Schemtic igrm o controlle genertion o quruple-component oule emulsions. Brek-up o these regulr roplet rrys in the intervls o perioic units cn le to co-encpsultion o two ierent roplets with precisely controlle numer n rtio. S4

5 S.. Controlle genertion o quruple-component oule emulsions contining two kins o roplets with ierent sizes The size o roplet cn e tune y chnging the low rtes n/or inner the inner imeter ( i ) o the collection tue o rop mkers. We illustrte the controllility o our microluiic evice over the size o co-encpsulte roplets y generting quruple-component oule emulsions tht contin smller re roplets n lrger lue roplets (Fig.S4). Here we use rop mkers with ierent i in rnch chnnels to generte the smller re roplets n lrger lue roplets (Fig.). Also, the numer n rtio o these two ierent roplets cn e controlle y justment o the low rte. The numer o smller re roplets increses rom 0 to 5 while tht o lrger lue roplets keeps ixe t, resulte in controlle rtio o smller re roplets to lrger lue roplets chnging rom : to 5:, s shown in Fig.S4()-(). For genertion o quruple-component oule emulsions contining smller re roplets n lrger lue roplets, the inner luis F - n F - re enzyl enzote oil phse with n without Sun re respectively, F -, F -, F -C n F -E re queous phses, n F -M is soyen oil phse, s shown in Fig.. The vrition rnges or the low rtes re - 0~50 μl h -, - 50~750 μl h -, μl h -, μl h -, -C 500~800 μl h -, -E 00 μl h - n -M 4000~5000 μl h -. Fig.S4 Opticl microgrphs showing controlle prouction o quruple-component oule emulsions with the numer o smller re roplets incresing rom 0 to 5 n tht o lrger lue roplets eing ixe t. Scle r is 00 μm. S.. Inepenent control o connector n liqui extrctor over the size o the outer roplets in quruple-component oule emulsions S5

6 Besies oning the min chnnel with rnch chnnels, nother importnt unctionlity o connector lies in the justment o intervl istnce etween roplets in the min chnnel. When we keep the other luis ixe, the only increse in the injecting low rte o connector les to n incresing istnce etween every two roplets. To relize the encpsultion o two ierent roplets, it requires more continuous lui tht crries these ierent roplets to e snwiche, resulting in n incresing size o the outer roplets (Fig.S5()). Similrly, the increse in the low rte o liqui extrctor les to ecresing istnce etween every two roplets. So, less continuous lui nees to e snwiche to relize the encpsultion o two ierent roplets, which results in ecresing size o the outer roplets (Fig.S5()). Note tht, ecuse the inner roplets re irst generte in rnch chnnels, chnge o the low rtes in min chnnel will not ect the size o these inner roplets. Thereore, with comine unctionlities o connector n liqui extrctor, the size o outer roplets in quruple-component oule emulsions cn e inepenently n precisely controlle. Fig.S5 Schemtic igrm showing the low-rte-epenent control over the size o outer roplets y connector () n liqui extrctor (). The istnce etween roplets cn e juste y chnging the low rtes o connector n liqui extrctor so s to tune the size o outer roplets. S.4. evelopment o preiction equtions or inner structures o quruple-component oule emulsions For quruple-component oule emulsions, the totl numer (N) o oth inner re n trnsprent roplets is controlle y mtching the ormtion rtes o inner re ( - ) n S

7 trnsprent roplets ( - ) with tht o the outer roplets ( ). Accoring to mss conservtion, the ormtion rte o ech roplet cn e escrie s ollows: (S) (S) (S) Where i-j is the low rte o lui F i-j ; is the sum o -, -, -, -, -C n -E ; i is the roplet imeter (s shown in Fig.). For ixe evice imensions ( i ) n solution conitions, the roplet imeter ( i ) in coxil low epens on the velocity o the surrouning low in the ripping regime [S], which cn e escrie s ollows: (S4) (S5) M (S) Where coeicients i n i re the slopes n intercepts o these liner reltions [S]. Bse on Equtions (S) to (S), we cn quntittively preict the vlue o N using the ollowing eqution: ( ) ( ) ( ) ( ) ( ) ( ) M N (S7) The rtio (R ) o inner re roplets to trnsprent roplets is etermine y the numer o inner re n trnsprent roplets encpsulte in the oule emulsions, which cn e seprtely controlle y justing - n -. Thereore, we cn quntittively etermine the vlue o R using the ollowing eqution: R (S8) S7

8 For ixe evices n emulsion systems, i, i n i re constnt, so these equtions or preiction o inner structure llow us to esign multicomponent multiple emulsions with esire internls through ine justment o the low rtes. Moreover, preiction equtions or higher-orer multicomponent multiple emulsions cn e urther evelope ccoring to the mss conservtion o ech lui phse. Reerences [S] L. Y. Chu, A. S. Ut, R. K. Shh, J. W. Kim n. A. Weitz, Angew. Chem. Int. E., 007, 4, S8

Instantaneous Rate of Change of at a :

Instantaneous Rate of Change of at a : AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim

More information

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

More information

Conservation Law. Chapter Goal. 6.2 Theory

Conservation Law. Chapter Goal. 6.2 Theory Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the

More information

AP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight

AP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight AP Clculus BC Review Applictions of Integrtion (Chpter Things to Know n Be Able to Do Fin the re between two curves by integrting with respect to x or y Fin volumes by pproximtions with cross sections:

More information

LECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for

LECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

Physics Lecture 14: MON 29 SEP

Physics Lecture 14: MON 29 SEP Physics 2113 Physics 2113 Lecture 14: MON 29 SEP CH25: Cpcitnce Von Kleist ws le to store electricity in the jr. Unknowingly, he h ctully invente novel evice to store potentil ifference. The wter in the

More information

Sturm-Liouville Theory

Sturm-Liouville Theory LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

More information

CLASS XII PHYSICS. (a) 30 cm, 60 cm (b) 20 cm, 30 cm (c) 15 cm, 20 cm (d) 12 cm, 15 cm. where

CLASS XII PHYSICS. (a) 30 cm, 60 cm (b) 20 cm, 30 cm (c) 15 cm, 20 cm (d) 12 cm, 15 cm. where PHYSICS combintion o two thin lenses with ocl lengths n respectively orms n imge o istnt object t istnce cm when lenses re in contct. The position o this imge shits by cm towrs the combintion when two

More information

4.5 THE FUNDAMENTAL THEOREM OF CALCULUS

4.5 THE FUNDAMENTAL THEOREM OF CALCULUS 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere

More information

x dx does exist, what does the answer look like? What does the answer to

x dx does exist, what does the answer look like? What does the answer to Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl

More information

Answers and Solutions to (Some Even Numbered) Suggested Exercises in Chapter 11 of Grimaldi s Discrete and Combinatorial Mathematics

Answers and Solutions to (Some Even Numbered) Suggested Exercises in Chapter 11 of Grimaldi s Discrete and Combinatorial Mathematics Answers n Solutions to (Some Even Numere) Suggeste Exercises in Chpter 11 o Grimli s Discrete n Comintoril Mthemtics Section 11.1 11.1.4. κ(g) = 2. Let V e = {v : v hs even numer o 1 s} n V o = {v : v

More information

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction Ch. 4 Moleculr Rection Dynmics 1. Collision Theory. Diffusion-Controlle Rection Lecture 17 3. The Mteril Blnce Eqution 4. Trnsition Stte Theory: The Eyring Eqution 5. Trnsition Stte Theory: Thermoynmic

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

polyimide Spray-coated ZrP/epoxy film Spray-coated ZrP/epoxy film glass

polyimide Spray-coated ZrP/epoxy film Spray-coated ZrP/epoxy film glass c d e polyimide Spry-coted ZrP/epoxy film glss Spry-coted ZrP/epoxy film f g Supplementry Figure 1. Opticl microscopy of smectic ( = 0.044) α-zrp/epoxy films., Trnsmission opticl microscopy (TOM) of smectic

More information

Conservation Law. Chapter Goal. 5.2 Theory

Conservation Law. Chapter Goal. 5.2 Theory Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very

More information

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

More information

Homework Assignment 5 Solution Set

Homework Assignment 5 Solution Set Homework Assignment 5 Solution Set PHYCS 44 3 Februry, 4 Problem Griffiths 3.8 The first imge chrge gurntees potentil of zero on the surfce. The secon imge chrge won t chnge the contribution to the potentil

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

0.1 THE REAL NUMBER LINE AND ORDER

0.1 THE REAL NUMBER LINE AND ORDER 6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.

More information

a n = 1 58 a n+1 1 = 57a n + 1 a n = 56(a n 1) 57 so 0 a n+1 1, and the required result is true, by induction.

a n = 1 58 a n+1 1 = 57a n + 1 a n = 56(a n 1) 57 so 0 a n+1 1, and the required result is true, by induction. MAS221(216-17) Exm Solutions 1. (i) A is () bounded bove if there exists K R so tht K for ll A ; (b) it is bounded below if there exists L R so tht L for ll A. e.g. the set { n; n N} is bounded bove (by

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the

More information

BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

More information

Applied. Grade 9 Assessment of Mathematics. Released assessment Questions

Applied. Grade 9 Assessment of Mathematics. Released assessment Questions Applie Gre 9 Assessment of Mthemtics 21 Relese ssessment Questions Recor your nswers to the multiple-choice questions on the Stuent Answer Sheet (21, Applie). Plese note: The formt of this booklet is ifferent

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

Review of Gaussian Quadrature method

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

More information

Topics Covered AP Calculus AB

Topics Covered AP Calculus AB Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

More information

If we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as

If we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is well-efine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x

More information

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION

SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION Physics 8.06 Apr, 2008 SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION c R. L. Jffe 2002 The WKB connection formuls llow one to continue semiclssicl solutions from n

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

MATH , Calculus 2, Fall 2018

MATH , Calculus 2, Fall 2018 MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly

More information

ES.181A Topic 8 Notes Jeremy Orloff

ES.181A Topic 8 Notes Jeremy Orloff ES.8A Topic 8 Notes Jeremy Orloff 8 Integrtion: u-substitution, trig-substitution 8. Integrtion techniques Only prctice will mke perfect. These techniques re importnt, but not the intellectul hert of the

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

Chapter 6 Techniques of Integration

Chapter 6 Techniques of Integration MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

New data structures to reduce data size and search time

New data structures to reduce data size and search time New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute

More information

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS F. Tkeo 1 nd M. Sk 1 Hchinohe Ntionl College of Technology, Hchinohe, Jpn; Tohoku University, Sendi, Jpn Abstrct:

More information

Chapter 3 Single Random Variables and Probability Distributions (Part 2)

Chapter 3 Single Random Variables and Probability Distributions (Part 2) Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their

More information

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

BME 207 Introduction to Biomechanics Spring 2018

BME 207 Introduction to Biomechanics Spring 2018 April 6, 28 UNIVERSITY O RHODE ISAND Deprtment of Electricl, Computer nd Biomedicl Engineering BME 27 Introduction to Biomechnics Spring 28 Homework 8 Prolem 14.6 in the textook. In ddition to prts -e,

More information

Part I: Basic Concepts of Thermodynamics

Part I: Basic Concepts of Thermodynamics Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION doi:.38/nture8499 doi:.38/nture8499 5 6 5 4.5 Firing rte (Hz) -67-65 -66-6 -58 V m (mv) -7-67 -68-66 -64 c Thet power (mv ) -73-69 -7-7 -7.5.8 3....9.9.4.6.6. 9 8 9 8 9 8 9 8 9 8 Supplementry Figure Firing

More information

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

More information

The Thermodynamics of Aqueous Electrolyte Solutions

The Thermodynamics of Aqueous Electrolyte Solutions 18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong

More information

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step

More information

Necessary and sufficient conditions for some two variable orthogonal designs in order 44

Necessary and sufficient conditions for some two variable orthogonal designs in order 44 University of Wollongong Reserch Online Fculty of Informtics - Ppers (Archive) Fculty of Engineering n Informtion Sciences 1998 Necessry n sufficient conitions for some two vrile orthogonl esigns in orer

More information

First Midterm Examination

First Midterm Examination 24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet

More information

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

More information

Last Time emphasis on E-field. Potential of spherical conductor. Quick quiz. Connected spheres. Varying E-fields on conductor.

Last Time emphasis on E-field. Potential of spherical conductor. Quick quiz. Connected spheres. Varying E-fields on conductor. Lst Time emphsis on Efiel Electric flux through surfce Guss lw: Totl electric flux through close surfce proportionl to chrge enclose Q " E = E = 4$k e Q % o Chrge istribution on conuctors Chrge ccumultes

More information

Evaluating Definite Integrals. There are a few properties that you should remember in order to assist you in evaluating definite integrals.

Evaluating Definite Integrals. There are a few properties that you should remember in order to assist you in evaluating definite integrals. Evluting Definite Integrls There re few properties tht you should rememer in order to ssist you in evluting definite integrls. f x dx= ; where k is ny rel constnt k f x dx= k f x dx ± = ± f x g x dx f

More information

Applications of Bernoulli s theorem. Lecture - 7

Applications of Bernoulli s theorem. Lecture - 7 Applictions of Bernoulli s theorem Lecture - 7 Prcticl Applictions of Bernoulli s Theorem The Bernoulli eqution cn be pplied to gret mny situtions not just the pipe flow we hve been considering up to now.

More information

Measuring Electron Work Function in Metal

Measuring Electron Work Function in Metal n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd

More information

The practical version

The practical version Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

More information

The development of nanoscale morphology in polymer:fullerene. photovoltaic blends during solvent casting

The development of nanoscale morphology in polymer:fullerene. photovoltaic blends during solvent casting Supplementry informtion Supplementry Mteril (ES) for Soft Mtter The development of nnoscle morphology in polymer:fullerene photovoltic lends during solvent csting To Wng, * Aln D. F. Dunr, Pul A. Stniec,

More information

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION EMF Notes 9; Electromgnetic nduction EECTOMAGNETC NDUCTON (Y&F Chpters 3, 3; Ohnin Chpter 3) These notes cover: Motionl emf nd the electric genertor Electromgnetic nduction nd Frdy s w enz s w nduced electric

More information

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x  0 : Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given

More information

Harman Outline 1A1 Integral Calculus CENG 5131

Harman Outline 1A1 Integral Calculus CENG 5131 Hrmn Outline 1A1 Integrl Clculus CENG 5131 September 5, 213 III. Review of Integrtion A.Bsic Definitions Hrmn Ch14,P642 Fundmentl Theorem of Clculus The fundmentl theorem of clculus shows the intimte reltionship

More information

4.4 Areas, Integrals and Antiderivatives

4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below . Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

More information

Math 142: Final Exam Formulas to Know

Math 142: Final Exam Formulas to Know Mth 4: Finl Exm Formuls to Know This ocument tells you every formul/strtegy tht you shoul know in orer to o well on your finl. Stuy it well! The helpful rules/formuls from the vrious review sheets my be

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

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

An Overview of Integration

An Overview of Integration An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Terminal Velocity and Raindrop Growth

Terminal Velocity and Raindrop Growth Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,

More information

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

More information

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims

More information

Version 001 HW#6 - Electromagnetic Induction arts (00224) 1 3 T

Version 001 HW#6 - Electromagnetic Induction arts (00224) 1 3 T Version 001 HW#6 - lectromgnetic Induction rts (00224) 1 This print-out should hve 12 questions. Multiple-choice questions my continue on the next column or pge find ll choices efore nswering. AP 1998

More information

Math Calculus with Analytic Geometry II

Math Calculus with Analytic Geometry II orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from

More information

Note 12. Introduction to Digital Control Systems

Note 12. Introduction to Digital Control Systems Note Introduction to Digitl Control Systems Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd . Introduction A digitl control system is one in which the

More information

1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang

1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang nonlinermc Fin solution root to nonliner lgebric eqution ()= Instructor: Nm Sun Wng Bckgroun In science n engineering, we oten encounter lgebric equtions where we wnt to in root(s) tht stisies given eqution

More information

Topics Review Fuel Conversion Efficiency Fuel Air Ratio Volumetric Efficiency Road Load Power Relationships between performance parameters

Topics Review Fuel Conversion Efficiency Fuel Air Ratio Volumetric Efficiency Road Load Power Relationships between performance parameters ME 410 Dy 5 Topics Reiew Fuel Conersion Eiciency Fuel Air Rtio Volumetric Eiciency Ro Lo Power Reltionships between perormnce prmeters Fuel Conersion Eiciency This is the rtio o power ctully prouce to

More information

FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

ELETROSTATICS Part II: BASICS

ELETROSTATICS Part II: BASICS GROWING WITH ONPTS: Physics LTROSTTIS Prt II: SIS Presence of chrge on ny oject cretes n electrosttic fiel roun it n in turn n electricl potentil is experience roun the oject. This phenomenon hs foun ppliction

More information

Final Exam - Review MATH Spring 2017

Final Exam - Review MATH Spring 2017 Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

INTRODUCTION TO LINEAR ALGEBRA

INTRODUCTION TO LINEAR ALGEBRA ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

MATH SS124 Sec 39 Concepts summary with examples

MATH SS124 Sec 39 Concepts summary with examples This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

More information

First Semester Review Calculus BC

First Semester Review Calculus BC First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

K + -Recognition Capsules with Squirting Release Mechanisms

K + -Recognition Capsules with Squirting Release Mechanisms Supplementary Material (ESI) for Chemical Communications K + -Recognition Capsules with Squirting Release Mechanisms Supplementary Material Zhuang Liu, Li Liu, Xiao-Jie Ju,* Rui Xie, Bao Zhang, and Liang-Yin

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorin Certificte of Euction 06 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Friy 4 November 06 Reing time: 9.00 m to 9.5 m (5 minutes) Writing

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

Lesson 8. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER)

Lesson 8. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER) Lesson 8 Thermomechnicl Mesurements for Energy Systems (MEN) Mesurements for Mechnicl Systems nd Production (MME) A.Y. 205-6 Zccri (ino ) Del Prete Mesurement of Mechnicl STAIN Strin mesurements re perhps

More information

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

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

More information

Torsion in Groups of Integral Triangles

Torsion in Groups of Integral Triangles Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,

More information

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length EMISTRY 104 elp Sheet #1 hem 103 Review (Text: h 6, h 7) Do topics pproprite for your lecture Prepred y Dr. Tony Jco http://www.chem.wisc.edu/res/clc (Resource pge) Nuggets: Electronegtivity (6.7), Bond

More information