CHAPTER TWO MULTIPLE INTEGRAL
|
|
|
- Tracey Barrett
- 5 years ago
- Views:
Transcription
1 CHAPTE TWO MULTIPLE INTEGAL Aft complting ths tutoils, stunts shoul b bl to: vlut th oubl intgl ov th givn ctngul gion fin th volum of th soli boun b th plns fin th of th gion boun b th cuvs ug oubl intgl fin th volum of th soli boun b th gphs ug oubl intgl fin th of th gion b ug oubl intgl in pol cooints chng th intgn fom Ct to pol cooints ov th givn gion G clcult f (,, V G fin th volum of th soli boun b th plns ug tipl intgl fin th volum of th soli boun b th givn sufcs ug tipl intgls in clinicl cooint fin th volum of th soli boun b th plns givn ug tipl intgls in sphicl cooint fin th mss of th lmin gion fin th mss n th cnt of mss of lmin gion boun b th givn gph n its nsit fin th cntoi fo th givn gion
2 Qustion Evlut th oubl intgl A ov th ctngul gion, :,. A u u u. Substitution Mtho: u u u Qustion Fin th volum of th soli boun b th pln,, n cooint pln. Solution Figu. ( Figu. (b
3 Volum of soli is givn b, V f (, A wh f (, : Thfo, V f (, A unit.
4 7 Qustion Evlut u u u Substitution Mtho: u u u
5 Qustion Lt b gion in th -pln n boun b,, Solution. Evlut A. Figu. A Qustion B ug th oubl intgl, fin th of th gion boun b th cuvs blow: 9,, Solution 9 Figu.
6 9 A, A A unit Qustion B ug th oubl intgl, fin th volum of th soli boun b ths gphs:,,, Solution Figu. ( Figu. (b
7 Volum, V f (, A A u u - simplif u u u - substitu u ( unit Substitution Mtho: u u u
8 Qustion 7 Sktch th gion boun b th gphs blow n fin th of th gion b ug oubl intgl in pol cooints :,,, Solution : A, A A = unit Qustion B chnging th intgn fom Ct to pol cooints, vlut : ( Solution :. - Substitution Mtho; u, u u u u
9 = (b Solution : Whn Whn ln ln ln tn ln sc tn ln sc tn ln sc sc
10 Qustion 9 Evlut th itt intgl blow : Solution : =
11 Qustion Clcult G V givn tht th gion G is, G, :,, : Solution : = Qustion Ug tipl intgl, fin th volum of th soli boun b th plns givn blow : ( Clin n plns =, + =. Ellips t th -pln, -intcpt t n -intcpt t. -limit : = - = -limit : -limit : = =. V= V G
12 - - B A A= with ius cicl of A O ug th substitution of A=
13 u B = Substitution mtho: u u u u Thfo, V V A B unit G (b =, =, =, + + =. Solution : Whn =, =.,, Whn =, = Whn =, =.,,.,, V V G + =
14 7 unit (c,, Solution : G V V
15 ( ( B A A= with ius cicl of A B = Substitution mtho: u u u u u
16 9 Thfo, unit B A V V G (,,, Solution : G V V unit - + =
17 A of cicl with ius A= B = u Substitution mtho: u u u u Thfo, V V A B unit G Qustion Evlut th itt intgl blow: (
18 (b ( (c ( ( (
19 Qustion Evlut th itt intgl blow: ( ( ( ( (b 7 ( ( ( 7 7
20 Qustion Clcult V G givn tht th gion G is fin s:,, :,, G (
21 Qustion Ug tipl intgl, fin th volum of th soli boun b th plns givn blow: (,,, Volum, V G A V unit ( ( ( ( ( ( (b Clin + = n plns =, + = =
22 Volum, V G A V ( ( ( ( ( ( Hlf of of cicl (smicicl with ius i.. Lt wh - - => cicl with ius
23 Thus, of th sh gion (i. smicicl is unit Altntiv mtho: Fo ( ( - - =
24 7 Thus, volum, unit V
25 Qustion Evlut th clinicl cooint intgls blow: ( 9 (b Fo A: Us substitution u u u u u A u u B u
26 9 Fo B: Thus, = A B Qustion 7 Fin th volum of th soli boun b th sufcs blow ug tipl intgls in clinicl cooint. (,, Volum, V - -
27 unit (b,, Volum, V ( ( - -
28 unit Qustion Evlut th sphicl cooint intgls blow: ( sc sc sc sc tn sc tn sc tn A B Fo A: Fo B: Substitut u tn u sc u sc u tn sc tn uu
29 tn Thus, sc = A B Qustion 9 Fin th volum of th soli boun b th plns givn blow: ( Sph =, con = /, n con = / Volum, V
30 Qustion Fin th mss of th lmin gion boun b th gph,,, n th nsit (,. = = Mss of th lmin gion (, A Qustion Fin th mss n th cnt of mss of lmin gion boun b th givn gph n nsit: (,, -is; (,
31 Mss of th lmin, m ( Cnt of mss = m M m M,, A M, ( 9 (
32 M (, A M ; m M m Thus, cnt of th mss is 9, 7. (b, cooints; (, Mss of th lmin, m = A Us oubl intgl in pol cooints; Thus, m
33 ( A M Us substitution u u ( u u u
34 7 A M Us substitution u u ( u u u Thus, cnt of mss = m M m M,,,,, Qustion Fin th cntoi fo th givn gions: ( Th tingl gion nclos b =, = n is.
35 Cntoi, A A A of, A of, A of = (( A A Thus, th cntoi is,,, =
36 9 (b Th gion boun b = n = A of gion 9 ( ( A A 9 ( - ( ( ( = (; ( ( = -, =
37 A 9 ( ( ( Thus, th cntoi is, 9 9, , 9 9,
DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3
DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl
Path (space curve) Osculating plane
Fo th cuilin motion of pticl in spc th fomuls did fo pln cuilin motion still lid. But th my b n infinit numb of nomls fo tngnt dwn to spc cu. Whn th t nd t ' unit ctos mod to sm oigin by kping thi ointtions
Homework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:
. Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo
Chapter 4 Circular and Curvilinear Motions
Chp 4 Cicul n Cuilin Moions H w consi picls moing no long sigh lin h cuilin moion. W fis su h cicul moion, spcil cs of cuilin moion. Anoh mpl w h l sui li is h pojcil..1 Cicul Moion Unifom Cicul Moion
MA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.
MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ
Case Study VI Answers PHA 5127 Fall 2006
Qustion. A ptint is givn 250 mg immit-rls thophyllin tblt (Tblt A). A wk ltr, th sm ptint is givn 250 mg sustin-rls thophyllin tblt (Tblt B). Th tblts follow on-comprtmntl mol n hv first-orr bsorption
sin sin 1 d r d Ae r 2
Diffction k f c f Th Huygn-Fnl Pincil tt: Evy unobtuct oint of vfont, t givn intnt, v ouc of hicl cony vlt (ith th m funcy tht of th imy v. Th mlitu of th oticl fil t ny oint byon i th uoition of ll th
Floating Point Number System -(1.3)
Floting Point Numbr Sstm -(.3). Floting Point Numbr Sstm: Comutrs rrsnt rl numbrs in loting oint numbr sstm: F,k,m,M 0. 3... k ;0, 0 i, i,...,k, m M. Nottions: th bs 0, k th numbr o igts in th bs xnsion
Floating Point Number System -(1.3)
Floting Point Numbr Sstm -(.3). Floting Point Numbr Sstm: Comutrs rrsnt rl numbrs in loting oint numbr sstm: F,k,m,M 0. 3... k ;0, 0 i, i,...,k, m M. Nottions: th bs 0, k th numbr o igits in th bs xnsion
y cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]
![y cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]](/thumbs/88/115680282.jpg "y cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]")
DIFF EQ - OPTION. Sol th iffrntial quation tan +, 0
D Properties and Measurement
APPENDIX D. Review of Alge, Geomet, nd Tigonomet A D Popetie nd Meuement D. Review of Alge, Geomet, nd Tigonomet Alge Popetie of Logitm Geomet Plne Anltic Geomet Solid Anltic Geomet Tigonomet Li of Function
Appendix D: Formulas, Properties and Measurements
Appendi D: Fomul, Popetie nd Meuement Review of Alge, Geomet, nd Tigonomet Unit of Meuement D. REVIEW OF ALGEBRA, GEOMETRY, AND TRIGONOMETRY Alge Popetie of Logitm Geomet Plne Anltic Geomet Solid Anltic
Homework: Study 6.2 #1, 3, 5, 7, 11, 15, 55, 57
Gols: 1. Undestnd volume s the sum of the es of n infinite nume of sufces. 2. Be le to identify: the ounded egion the efeence ectngle the sufce tht esults fom evolution of the ectngle ound n xis o foms
The angle between L and the z-axis is found from
Poblm 6 This is not a ifficult poblm but it is a al pain to tansf it fom pap into Mathca I won't giv it to you on th quiz, but know how to o it fo th xam Poblm 6 S Figu 6 Th magnitu of L is L an th z-componnt
Convergence tests for the cluster DFT calculations
Covgc ss o h clus DF clculos. Covgc wh spc o bss s. s clculos o bss s covgc hv b po usg h PBE ucol o 7 os gg h-b. A s o h Guss bss ss wh csg s usss hs b us clug h -G -G** - ++G(p). A l sc o. Å h c bw h
1 Using Integration to Find Arc Lengths and Surface Areas
Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s
+ r Position Velocity
1. The phee P tel in tight line with contnt peed of =100 m/. Fo the intnt hown, detemine the coeponding lue of,,,,, eltie to the fixed Ox coodinte tem. meued + + Poition Velocit e 80 e 45 o 113. 137 d
MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4
MATHEMATICS IV MARKS. If + + 6 + c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce
1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan
LOCUS 58 SOLVED EXAMPLES Empl Lt F n F th foci of n llips with ccntricit. For n point P on th llips, prov tht tn PF F tn PF F Assum th llips to, n lt P th point (, sin ). P(, sin ) F F F = (-, 0) F = (,
Solutions to Supplementary Problems
Solution to Supplmntay Poblm Chapt Solution. Fomula (.4): g d G + g : E ping th void atio: G d 2.7 9.8 0.56 (56%) 7 mg Fomula (.6): S Fomula (.40): g d E ping at contnt: S m G 0.56 0.5 0. (%) 2.7 + m E
Answers to test yourself questions
Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E
U>, and is negative. Electric Potential Energy
Electic Potentil Enegy Think of gvittionl potentil enegy. When the lock is moved veticlly up ginst gvity, the gvittionl foce does negtive wok (you do positive wok), nd the potentil enegy (U) inceses. When
Lecture 35. Diffraction and Aperture Antennas
ctu 35 Dictin nd ptu ntnns In this lctu u will ln: Dictin f lctmgntic ditin Gin nd ditin pttn f ptu ntnns C 303 Fll 005 Fhn Rn Cnll Univsit Dictin nd ptu ntnns ptu ntnn usull fs t (mtllic) sht with hl
1. The sphere P travels in a straight line with speed
1. The sphee P tels in stight line with speed = 10 m/s. Fo the instnt depicted, detemine the coesponding lues of,,,,, s mesued eltie to the fixed Oxy coodinte system. (/134) + 38.66 1.34 51.34 10sin 3.639
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
Aakash. For Class XII Studying / Passed Students. Physics, Chemistry & Mathematics
Aakash A UNIQUE PPRTUNITY T HELP YU FULFIL YUR DREAMS Fo Class XII Studying / Passd Studnts Physics, Chmisty & Mathmatics Rgistd ffic: Aakash Tow, 8, Pusa Road, Nw Dlhi-0005. Ph.: (0) 4763456 Fax: (0)
Thomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
Fluids & Bernoulli s Equation. Group Problems 9
Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going
Kinetics. Central Force Motion & Space Mechanics
Kintics Cntal Foc Motion & Spac Mcanics Outlin Cntal Foc Motion Obital Mcanics Exampls Cntal-Foc Motion If a paticl tavls un t influnc of a foc tat as a lin of action ict towas a fix point, tn t motion
8 - GRAVITATION Page 1
8 GAVITATION Pag 1 Intoduction Ptolmy, in scond cntuy, gav gocntic thoy of plantay motion in which th Eath is considd stationay at th cnt of th univs and all th stas and th plants including th Sun volving
PhysicsAndMathsTutor.com
PhsicsAMthsTuto.com 6. The hpeol H hs equtio, whee e costts. The lie L hs equtio m c, whee m c e costts. Leve lk () Give tht L H meet, show tht the -cooites of the poits of itesectio e the oots of the
Thomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
ORBITAL TO GEOCENTRIC EQUATORIAL COORDINATE SYSTEM TRANSFORMATION. x y z. x y z GEOCENTRIC EQUTORIAL TO ROTATING COORDINATE SYSTEM TRANSFORMATION
ORITL TO GEOCENTRIC EQUTORIL COORDINTE SYSTEM TRNSFORMTION z i i i = (coωcoω in Ωcoiinω) (in Ωcoω + coωcoiinω) iniinω ( coωinω in Ωcoi coω) ( in Ωinω + coωcoicoω) in icoω in Ωini coωini coi z o o o GEOCENTRIC
Trigonometry. Angle Measurement radians 180. Fundamental Identities. Right Angle Trigonometry. Trigonometric Functions. The Law of Sines sin A
Alge Geomet Aithmetic Opetions c c c c Exponents n Ricls x m x n x mn c c x m mn x x n c c c Geometic Fomls Fomls fo e A, cicmfeence C, n volme V: Tingle Cicle Secto of Cicle A h A A sin C s in ins h s
Theory of Spatial Problems
Chpt 7 ho of Sptil Polms 7. Diffntil tions of iliim (-D) Z Y X Inol si nknon stss componnts:. 7- 7. Stt of Stss t Point t n sfc ith otd noml N th sfc componnts ltd to (dtmind ) th 6 stss componnts X N
Boyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues
BocDPm 9 h d Ch 7.6: Compl Egvlus Elm Dffl Equos d Boud Vlu Poblms 9 h do b Wllm E. Boc d Rchd C. DPm 9 b Joh Wl & Sos Ic. W cosd g homogous ssm of fs od l quos wh cos l coffcs d hus h ssm c b w s ' A
π,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
AP Calculus AB Exam Review Sheet B - Session 1
AP Clcls AB Em Review Sheet B - Session Nme: AP 998 # Let e the nction given y e.. Find lim nd lim.. Find the solte minimm vle o. Jstiy tht yo nswe is n solte minimm. c. Wht is the nge o? d. Conside the
4.2 Boussinesq s Theory. Contents
00477 Pvement Stuctue 4. Stesses in Flexible vement Contents 4. Intoductions to concet of stess nd stin in continuum mechnics 4. Boussinesq s Theoy 4. Bumiste s Theoy 4.4 Thee Lye System Weekset Sung Chte
STATISTICAL MECHANICS OF DIATOMIC GASES
Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) SAISICAL MECHAICS OF DIAOMIC GASES - Fo monatomic gas whos molculs hav th dgs of fdom of tanslatoy motion th intnal u 3 ngy and th spcific
Solution of Math132 Exam1
Solution of Math3 Exam. ( %) (a) A bacteria has population at time t = an its rate of growth is (t 3 + t + ) bacteria per hour after t hours. What is the population after 4 hours? Suppose p(t) is the bacteria
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:
ELEC 351 Notes Set #18
Assignmnt #8 Poblm 9. Poblm 9.7 Poblm 9. Poblm 9.3 Poblm 9.4 LC 35 Nots St #8 Antnns gin nd fficincy Antnns dipol ntnn Hlf wv dipol Fiis tnsmission qution Fiis tnsmission qution Do this ssignmnt by Novmb
ME 425: Aerodynamics
ME 5: Aeodynamics D ABM Toufique Hasan Pofesso Depatment of Mechanical Engineeing, BUET Lectue- 8 Apil 7 teachebuetacbd/toufiquehasan/ toufiquehasan@mebuetacbd ME5: Aeodynamics (Jan 7) Flow ove a stationay
SOLUTIONS TO CONCEPTS CHAPTER 11
SLUTINS T NEPTS HPTE. Gvittionl fce of ttction, F.7 0 0 0.7 0 7 N (0.). To clculte the gvittionl fce on t unline due to othe ouse. F D G 4 ( / ) 8G E F I F G ( / ) G ( / ) G 4G 4 D F F G ( / ) G esultnt
CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.
CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.
This immediately suggests an inverse-square law for a "piece" of current along the line.
Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line
Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I
Exm Stuy Guie Mth 26 - Clulus II, Fll 205 The following is list of importnt onepts from eh setion tht will be teste on exm. This is not omplete list of the mteril tht you shoul know for the ourse, but
International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September ISSN
Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- bstct: Jcbin intgl nd Stbility f th quilibium psitin f th cnt f mss f n xtnsibl cbl cnnctd stllits systm in th lliptic bit. Vijy Kum ssistnt
Neural Networks The ADALINE
Lat Lctu Summay Intouction to ua to Bioogica uon Atificia uon McCuoch an itt LU Ronbatt cton Aan Bnaino, a@i.it.ut.t Machin Laning, 9/ ua to h ADALI M A C H I L A R I G 9 / cton Limitation cton aning u
WELSH JOINT EDUCATION COMMITTEE CYD-BWYLLGOR ADDYSG CYMRU MATHEMATICS. FORMULA BOOKLET (New Specification)
WELSH JOINT EDUCATION COMMITTEE CYD-BWYLLGOR ADDYSG CYMRU Gl Ciic o Eucio Avc Lvl/Avc Susii Tssgi Asg Giol So Uwch/Uwch Gol MATHEMATICS FORMULA BOOKLET Nw Spciicio Issu 004 Msuio Suc o sph 4π A o cuv suc
ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören
ME 522 PINCIPLES OF OBOTICS FIST MIDTEM EXAMINATION April 9, 202 Nm Lst Nm M. Kml Özgörn 2 4 60 40 40 0 80 250 USEFUL FOMULAS cos( ) cos cos sin sin sin( ) sin cos cos sin sin y/ r, cos x/ r, r 0 tn 2(
Boxing Blends Sub step 2.2 Focus: Beginning, Final, and Digraph Blends
Boxing Blends Sub step 2.2 Focus: Beginning, Final, and Digraph Blends Boxing Blends Game Instructions: (One Player) 1. Use the game board appropriate for your student. Cut out the boxes to where there
Period vs. Length of a Pendulum
Gaphcal Mtho n Phc Gaph Intptaton an Lnazaton Pat 1: Gaphng Tchnqu In Phc w u a vat of tool nclung wo, quaton, an gaph to mak mol of th moton of objct an th ntacton btwn objct n a tm. Gaph a on of th bt
Multiplying and Dividing Rational Expressions
Lesson Peview Pt - Wht You ll Len To multipl tionl epessions To divide tionl epessions nd Wh To find lon pments, s in Eecises 0 Multipling nd Dividing Rtionl Epessions Multipling Rtionl Epessions Check
Section 3: Antiderivatives of Formulas
Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin
QUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt of Computr n Informtion Sins CSCI 710 (Trnoff) Disrt Struturs TEST for Fll Smstr, 00 R this for strtin! This tst is los ook
(1) (Pre-Calc Review Set Problems # 44 and # 59) Show your work. (a) Given sec =5,andsin <0, find the exact values of the following functions.
() (Pre-Clc Review Set Problems # 44 n # 59) Show your work. () Given sec =5,nsin
St Andrew s Academy Mathematics Department Higher Mathematics VECTORS
St ndew s cdemy Mthemtics etment Highe Mthemtics VETORS St ndew's cdemy Mths et 0117 1 Vectos sics 1. = nd = () Sketch the vectos nd. () Sketch the vectos nd. (c) Given u = +, sketch the vecto u. (d) Given
Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.
Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,
CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find
BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,
Using integration tables
Using integrtion tbles Integrtion tbles re inclue in most mth tetbooks, n vilble on the Internet. Using them is nother wy to evlute integrls. Sometimes the use is strightforwr; sometimes it tkes severl
CHAPTER 5 CIRCULAR MOTION
CHAPTER 5 CIRCULAR MOTION and GRAVITATION 5.1 CENTRIPETAL FORCE It is known that if a paticl mos with constant spd in a cicula path of adius, it acquis a cntiptal acclation du to th chang in th diction
, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x,
Clculus for Businss nd Socil Scincs - Prof D Yun Finl Em Rviw vrsion 5/9/7 Chck wbsit for ny postd typos nd updts Pls rport ny typos This rviw sht contins summris of nw topics only (This rviw sht dos hv
dx was area under f ( x ) if ( ) 0
13. Line Integls Line integls e simil to single integl, f ( x) dx ws e unde f ( x ) if ( ) 0 Insted of integting ove n intevl [, ] (, ) f xy ds f x., we integte ove cuve, (in the xy-plne). **Figue - get
Ch 1.2: Solutions of Some Differential Equations
Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of
A Tutorial on Multiple Integrals (for Natural Sciences / Computer Sciences Tripos Part IA Maths)
A Tutoial on Multiple Integals (fo Natual Sciences / Compute Sciences Tipos Pat IA Maths) Coections to D Ian Rud (http://people.ds.cam.ac.uk/ia/contact.html) please. This tutoial gives some bief eamples
Using Potential Energy
Using Potentil Enegy You ve job poviing te engineeing elp o n citect in Coloo. You e cuently esigning cble tow to pull sies up ill so tey cn si own. e custoe woul lie te cble tow to pull sie upill t constnt
x ) dx dx x sec x over the interval (, ).
Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below.
oa Euatons Thoughout all of chapt 4, ou focus s on th machn tslf, thfo w wll only pfom a y smpl tatmnt of th ntwok n o to s a complt mol. W o that h, but alz that w wll tun to ths ssu n Chapt 9. So lt
Review of Mathematical Concepts
ENEE 322: Signls nd Systems view of Mthemticl Concepts This hndout contins ief eview of mthemticl concepts which e vitlly impotnt to ENEE 322: Signls nd Systems. Since this mteil is coveed in vious couses
Handout 7. Properties of Bloch States and Electron Statistics in Energy Bands
Hdout 7 Popts of Bloch Stts d Elcto Sttstcs Eg Bds I ths lctu ou wll l: Popts of Bloch fuctos Podc boud codtos fo Bloch fuctos Dst of stts -spc Elcto occupto sttstcs g bds ECE 407 Spg 009 Fh R Coll Uvst
Chapter I Vector Analysis
. Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw
C-Curves. An alternative to the use of hyperbolic decline curves S E R A F I M. Prepared by: Serafim Ltd. P. +44 (0)
An ltntiv to th us of hypolic dclin cuvs Ppd y: Sfim Ltd S E R A F I M info@sfimltd.com P. +44 (02890 4206 www.sfimltd.com Contnts Contnts... i Intoduction... Initil ssumptions... Solving fo cumultiv...
Additional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
11.2. Area of a Circle. Lesson Objective. Derive the formula for the area of a circle.
11.2 Aea of a Cicle Lesson Objective Use fomulas to calculate the aeas of cicles, semicicles, and quadants. Lean Deive the fomula fo the aea of a cicle. A diamete divides a cicle of adius into 2 semicicles.
Who is this Great Team? Nickname. Strangest Gift/Friend. Hometown. Best Teacher. Hobby. Travel Destination. 8 G People, Places & Possibilities
Who i thi Gt Tm? Exi Sh th foowing i of infomtion bot of with o tb o tm mt. Yo o not hv to wit n of it own. Yo wi b givn on 5 mint to omih thi tk. Stngt Gift/Fin Niknm Homtown Bt Th Hobb Tv Dtintion Robt
Electric Field F E. q Q R Q. ˆ 4 r r - - Electric field intensity depends on the medium! origin
1 1 Electic Field + + q F Q R oigin E 0 0 F E ˆ E 4 4 R q Q R Q - - Electic field intensity depends on the medium! Electic Flux Density We intoduce new vecto field D independent of medium. D E So, electic
ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:
LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6
Prelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours
Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd - hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn
EXPANSION OF LIQUIDS
EXPNSION OF LIQUIDS. block of woo is flotin on wte t C with cetin volume V bove wte level. The tempetue of wte is slowly ise fom C to C. How the volume V chne with the ise of tempetue ) V will emin unchne
STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0
STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed
ESCI 341 Atmospheric Thermodynamics Lesson 16 Pseudoadiabatic Processes Dr. DeCaria
ESCI 34 Atmohi hmoynami on 6 Puoaiabati Po D DCaia fn: Man, A an FE obitaill, 97: A omaion of th uialnt otntial tmatu an th tati ngy, J Atmo Si, 7, 37-39 Btt, AK, 974: Futh ommnt on A omaion of th uialnt
Chapter 28 Sources of Magnetic Field
Chpte 8 Souces of Mgnetic Field - Mgnetic Field of Moving Chge - Mgnetic Field of Cuent Element - Mgnetic Field of Stight Cuent-Cying Conducto - Foce Between Pllel Conductos - Mgnetic Field of Cicul Cuent
Classical Electrodynamics
Fist Look t Quntu hysics Cssic Eectoynics Chpte gnetosttics Fy s Lw Qusi-Sttic Fies Cssic Eectoynics of. Y. F. Chen Contents Fist Look t Quntu hysics. The etionship between eectic fie n gnetic fie. iot
4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
![4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.](/thumbs/86/93688166.jpg "4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.")
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
Algebra Based Physics. Gravitational Force. PSI Honors universal gravitation presentation Update Fall 2016.notebookNovember 10, 2016
Newton's Lw of Univesl Gvittion Gvittionl Foce lick on the topic to go to tht section Gvittionl Field lgeb sed Physics Newton's Lw of Univesl Gvittion Sufce Gvity Gvittionl Field in Spce Keple's Thid Lw
Much that has already been said about changes of variable relates to transformations between different coordinate systems.
MULTIPLE INTEGRLS I P Calculus Cooinate Sstems Much that has alea been sai about changes of vaiable elates to tansfomations between iffeent cooinate sstems. The main cooinate sstems use in the solution
Unit 5. Integration techniques
18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd
On the Eötvös effect
On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion
Chapter 7 Dynamic stability analysis I Equations of motion and estimation of stability derivatives - 4 Lecture 25 Topics
Chapt 7 Dynamic stability analysis I Equations of motion an stimation of stability ivativs - 4 ctu 5 opics 7.8 Expssions fo changs in aoynamic an populsiv focs an momnts 7.8.1 Simplifi xpssions fo changs
PhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com PhysicsAdMthsTuto.com Jue 009 7. () Sketch the gph of y, whee >, showig the coodites of the poits whee the gph meets the es. () Leve lk () Solve, >. (c) Fid the set of vlues of fo
MULTIPLE INTEGRALS. A double integral of a positive function is a volume, which is the limit of sums of volumes of rectangular columns.
5 MULTIPL INTGALS A double integrl of positive function is volume, which is the limit of sums of volumes of rectngulr columns. In this chpter we etend the ide of definite integrl to double nd triple integrls
Mid Year Examination F.4 Mathematics Module 1 (Calculus & Statistics) Suggested Solutions
Mid Ya Eamination 3 F. Matmatics Modul (Calculus & Statistics) Suggstd Solutions Ma pp-: 3 maks - Ma pp- fo ac qustion: mak. - Sam typ of pp- would not b countd twic fom wol pap. - In any cas, no pp maks
Lecture 11: Potential Gradient and Capacitor Review:
Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of
SPECIALIST MATHEMATICS
Victorin Certificte of Euction 08 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Friy 9 November 08 Reing time: 9.00 m to 9.5 m (5 minutes) Writing
Chapter 25 Electric Potential
Chpte 5 lectic Potentil consevtive foces -> potentil enegy - Wht is consevtive foce? lectic potentil = U / : the potentil enegy U pe unit chge is function of the position in spce Gol:. estblish the eltionship
Applied Aerodynamics
Applied Aeodynamics Def: Mach Numbe (M), M a atio of flow velocity to the speed of sound Compessibility Effects Def: eynolds Numbe (e), e ρ c µ atio of inetial foces to viscous foces iscous Effects If
Electric Potential. and Equipotentials
Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil