AREA OF A SURFACE OF REVOLUTION
|
|
|
- Megan Stokes
- 2 years ago
- Views:
Transcription
1 AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7. nd 7.. We wnt to define the re of surfce of revolution in such w tht it corresponds to our intuition. If the surfce re is A, we cn imgine tht pinting the surfce would require the sme mount of pint s does flt region with re A. Let s strt with some simple surfces. The lterl surfce re of circulr clinder with rdius r nd height h is tken to be A rh becuse we cn imgine cutting the clinder nd unrolling it (s in Figure ) to obtin rectngle with dimensions r nd h. Likewise, we cn tke circulr cone with bse rdius r nd slnt height l, cut it long the dshed line in Figure, nd fltten it to form sector of circle with rdius l nd centrl ngle. We know tht, in generl, the re of sector of circle with rdius l nd ngle is l (see Eercise 67 in Section 6.) nd so in this cse it is rl A l l r l rl FIGURE Therefore, we define the lterl surfce re of cone to be A rl. πr cut l r l FIGURE l Wht bout more complicted surfces of revolution? If we follow the strteg we used with rc length, we cn pproimte the originl curve b polgon. When this polgon is rotted bout n is, it cretes simpler surfce whose surfce re pproimtes the ctul surfce re. B tking limit, we cn determine the ect surfce re. The pproimting surfce, then, consists of number of bnds, ech formed b rotting line segment bout n is. To find the surfce re, ech of these bnds cn be considered portion of circulr cone, s shown in Figure. The re of the bnd (or frustum of cone) with slnt height l nd upper nd lower rdii r nd r is found b subtrcting the res of two cones: r A r l l r l r r l r l r l From similr tringles we hve l l l r r FIGURE which gives r l r l r l or r r l r l Putting this in Eqution, we get or A r l r l Thomson Brooks-Cole copright 7 A rl where r r r is the verge rdius of the bnd.
2 AREA OF A SURFACE OF REVOLUTION =ƒ b () Surfce of revolution Now we ppl this formul to our strteg. Consider the surfce shown in Figure, which is obtined b rotting the curve f, b, bout the -is, where f is positive nd hs continuous derivtive. In order to define its surfce re, we divide the intervl, b into n subintervls with endpoints,,..., n nd equl width, s we did in determining rc length. If i f i, then the point P i i, i lies on the curve. The prt of the surfce between i nd i is pproimted b tking the line segment P i P i nd rotting it bout the -is. The result is bnd with slnt height l P ip i nd verge rdius r i i so, b Formul, its surfce re is P i P i- P i P n i i P ip i b (b) Approimting bnd FIGURE As in the proof of Theorem 7.., we hve P ip i s f i* where i * is some number in i, i. When is smll, we hve i f i f i * nd lso i f i f i *, since f is continuous. Therefore i i P ip i f i * s f i * nd so n pproimtion to wht we think of s the re of the complete surfce of revolution is n i f i * s f i * This pproimtion ppers to become better s n l nd, recognizing () s Riemnn sum for the function t f s f, we hve lim n l n i f i * s f i * b Therefore, in the cse where f is positive nd hs continuous derivtive, we define the surfce re of the surfce obtined b rotting the curve f, b, bout the -is s f s f S b f s f With the Leibniz nottion for derivtives, this formul becomes 5 S b d If the curve is described s t, c d, then the formul for surfce re becomes Thomson Brooks-Cole copright 7 6 S d c d d nd both Formuls 5 nd 6 cn be summrized smbolicll, using the nottion for rc
3 AREA OF A SURFACE OF REVOLUTION length given in Section 7., s 7 S ds For rottion bout the -is, the surfce re formul becomes 8 S ds where, s before, we cn use either ds d or ds d d These formuls cn be remembered b thinking of or s the circumference of circle trced out b the point, on the curve s it is rotted bout the -is or -is, respectivel (see Figure 5). (, ) (, ) FIGURE 5 circumference=π () Rottion bout -is: S=j π ds circumference=π (b) Rottion bout -is: S=j π ds EXAMPLE The curve s,, is n rc of the circle. Find the re of the surfce obtined b rotting this rc bout the -is. (The surfce is portion of sphere of rdius. See Figure 6.) SOLUTION We hve d s nd so, b Formul 5, the surfce re is S d Thomson Brooks-Cole copright 7 FIGURE 6 Figure 6 shows the portion of the sphere whose surfce re is computed in Emple. s s s 8
4 AREA OF A SURFACE OF REVOLUTION Figure 7 shows the surfce of revolution whose re is computed in Emple. EXAMPLE The rc of the prbol from, to, is rotted bout the -is. Find the re of the resulting surfce. SOLUTION Using (, ) = we hve, from Formul 8, nd d FIGURE 7 S ds d s Substituting u, we hve du 8. Remembering to chnge the limits of integrtion, we hve S 7 5 su du [ u ] 5 7 As check on our nswer to Emple, notice from Figure 7 tht the surfce re should be close to tht of circulr clinder with the sme height nd rdius hlfw between the upper nd lower rdius of the surfce:. We computed tht the surfce re ws (7s7 5s5).85 6 which seems resonble. Alterntivel, the surfce re should be slightl lrger thn the re of frustum of cone with the sme top nd bottom edges. From Eqution, this is..5(s) 9.8 SOLUTION we hve Using S ds 6 s (7s7 5s5) nd d d d s s d s d su du (7s7 5s5) (where u ) (s in Solution ) Thomson Brooks-Cole copright 7 Another method: Use Formul 6 with ln. EXAMPLE Find the re of the surfce generted b rotting the curve e,,bout the -is. SOLUTION Using Formul 5 with e nd d e
5 AREA OF A SURFACE OF REVOLUTION 5 we hve S d e s e e s u du sec d (where u e ) tn e (where u tn nd ) Or use Formul in the Tble of Integrls. [sec tn ln sec tn ] (b Emple 8 in Section 6.) [sec tn lnsec tn s ln(s )] Since tn e, we hve sec tn e nd S [es e ln(e s e ) s ln(s )] EXERCISES A Click here for nswers. Set up, but do not evlute, n integrl for the re of the surfce obtined b rotting the curve bout the given is.. ln, ; -is. sin, ; -is. sec, ; -is. e, ; -is S Click here for solutions. 7 Use Simpson s Rule with n to pproimte the re of the surfce obtined b rotting the curve bout the -is. Compre our nswer with the vlue of the integrl produced b our clcultor. 7. ln, 8. s, 9. sec,. s e, 5 Find the re of the surfce obtined b rotting the curve bout the -is. 5., , 7. s, 8. cos, 9. cosh,., ,., CAS CAS Use either CAS or tble of integrls to find the ect re of the surfce obtined b rotting the given curve bout the -is..,. s, Use CAS to find the ect re of the surfce obtined b rotting the curve bout the -is. If our CAS hs trouble evluting the integrl, epress the surfce re s n integrl in the other vrible..,. ln, Thomson Brooks-Cole copright 7 6 The given curve is rotted bout the -is. Find the re of the resulting surfce.. s,., 5. s, 6. cosh, 5. () If, find the re of the surfce generted b rotting the loop of the curve bout the -is. (b) Find the surfce re if the loop is rotted bout the -is. 6. A group of engineers is building prbolic stellite dish whose shpe will be formed b rotting the curve bout the -is. If the dish is to hve -ft dimeter nd mimum depth of ft, find the vlue of nd the surfce re of the dish.
6 6 AREA OF A SURFACE OF REVOLUTION CAS 7. The ellipse b b is rotted bout the -is to form surfce clled n ellipsoid. Find the surfce re of this ellipsoid. 8. Find the surfce re of the torus in Eercise in Section If the curve f, b, is rotted bout the horizontl line c, where f c, find formul for the re of the resulting surfce.. Use the result of Eercise 9 to set up n integrl to find the re of the surfce generted b rotting the curve s,,bout the line. Then use CAS to evlute the integrl.. Find the re of the surfce obtined b rotting the circle r bout the line r.. Show tht the surfce re of zone of sphere tht lies between two prllel plnes is S dh, where d is the dimeter of the sphere nd h is the distnce between the plnes. (Notice tht S depends onl on the distnce between the plnes nd not on their loction, provided tht both plnes intersect the sphere.). Formul is vlid onl when f. Show tht when f is not necessril positive, the formul for surfce re becomes S b. Let L be the length of the curve f, b, where f is positive nd hs continuous derivtive. Let S f be the surfce re generted b rotting the curve bout the -is. If c is positive constnt, define t f c nd let S t be the corresponding surfce re generted b the curve t, b. Epress in terms of nd L. S t f s f S f Thomson Brooks-Cole copright 7
7 AREA OF A SURFACE OF REVOLUTION 7 ANSWERS.. S ln s Click here for solutions. s sec tn 5. (5s5 )7 7. (7s7 7s7)6 9. [ e e ].. (5s5 s) [ ln(s7 ) ln(s ) s7 s] 6[ln(s ) s]. 5. () (b) 56s 5 7. [b b sin (s b )s b ] 9. b f s f. r Thomson Brooks-Cole copright 7
8 8 AREA OF A SURFACE OF REVOLUTION SOLUTIONS. =ln ds = +(d/) = +(/) S = π(ln ) +(/) [b (7)]. =sec ds = +(d/) = +(sec tn ) S = π/ π +(sec tn ) [b (8)] 5. = =.So S = π +( ) =π = π 6 5 udu= π 8 +9 [u =+9, du =6 ] 5 u/ = π = +(d/) =+[/( )] =+/(). So S = 9 =π π + d + / 9 = π 9 = π 8 ( +)/ 9 + =π = π =cosh +(d/) =+sinh =cosh.so S =π cosh cosh =π ( + cosh ) = π + sinh = π + sinh or π + e e. = + / /d = + / () = + +(/d) =+ + = +.So S =π + d =π + =π + = π. = = +(/d) =+9.So S =π +(/d) d =π +9 d = π +9 6 d 6 +9 / 5 5 = π 8 = π 7 5. = /d = ( ) / ( ) = / +(/d) =+ S = / = + = π / d =π d =π / =π = π. Note tht this is Thomson Brooks-Cole copright 7 the surfce re of sphere of rdius, nd the length of the intervl =to = / is the length of the intervl = to =.
9 AREA OF A SURFACE OF REVOLUTION 9 7. =ln d/ =/ +(d/) =+/ S = π ln +/. Let f() =ln +/.Sincen =, = = 5. Then S S =π /5 [f() + f(.) + f(.) + +f(.6) + f(.8) + f()] The vlue of the integrl produced b clcultor is 9.6 (to si deciml plces). 9. =sec d/ =sec tn +(d/) =+sec tn S = π/ π sec +sec tn.letf() =sec +sec tn. Since n =, = π/ = π.then S S =π π/ π f() + f +f π + +f 8π +f The vlue of the integrl produced b clcultor is (to si deciml plces). 9π π + f =/ ds = +(d/) = +( / ) = +/ S = π + =π + u + =π u du [u =, du =] +u = π du +u = π +ln u + +u u u = π 7 +ln ln = π 7 +ln = nd = nd. S = π +( ) =π +u du [u 6 =, du =6] = π +u du π = [or use CAS] u +u + ln u + +u = π + ln + = π 6 + ln + 5. Since >,thecurve = ( ) onl hs points with. ( ( ).) The curve is smmetric bout the -is (since the eqution is unchnged when is replced b ). =when =or, so the curve s loop etends from =to =. Thomson Brooks-Cole copright 7 d ( )= d [( ) ] 6 d = ( )( ) + ( ) d ( )[ + ] = 6 d = ( ) ( ) = ( ) ( ) 6 6 ( ) the lst frction is / = ( )
10 AREA OF A SURFACE OF REVOLUTION + d = = = ( +) = for 6=. () S = = π = π ds =π ( + ) = π ( ) + ( )( +) =π 6 + = π ( + )= π = π. Note tht we hve rotted the top hlf of the loop bout the -is. This genertes the full surfce. (b) We must rotte the full loop bout the -is, so we get double the re obtined b rotting the top hlf of the loop: S = π ds=π = + = π / ( +) = π ( / + / ) = π / / = π 5/ / = π + 6 = π 5 8 = 56π (d/) = = d b b = b + d =+ b = b + = b + b / = b + b b b ( / ) b b = + b = b ( ) The ellipsoid s surfce re is twice the re generted b rotting the first qudrnt portion of the ellipse bout the -is. Thus, S = π + d b ( =π b ) = πb ( b ) = πb b u du b [u = b ] = πb b u u + u b sin πb b = b ( b )+ b sin =π b + b sin b b Thomson Brooks-Cole copright 7 9. The nlogue of f( i ) in the derivtion of () is now c f( i ), so S = lim n n i= π[c f( i )] +[f ( i )] = b π[c f()] +[f ()].
11 AREA OF A SURFACE OF REVOLUTION. For the upper semicircle, f() = r, f () = / r. The surfce re generted is r S = π r r + =π r r =π r r r r For the lower semicircle, f() = r nd f () = Thus, the totl re is S = S + S =8π r r r r r,sos =π r r =8π r sin r r r r r r r + r. =8πr π =π r.. In the derivtion of (), we computed tpicl contribution to the surfce re to be π i + i P i P i, the re of frustum of cone. When f() is not necessril positive, the pproimtions i = f( i ) f( i ) nd i = f( i ) f( i ) must be replced b i = f( i ) f( i ) nd i = f( i ) f( i ). Thus, π i + i P i P i π f( i ) +[f ( i )]. Continuing with the rest of the derivtion s before, we obtin S = b π f() +[f ()]. Thomson Brooks-Cole copright 7
Double Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
Arc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
Volumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using
5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued
So there are two points of intersection, one being x = 0, y = 0 2 = 0 and the other being x = 2, y = 2 2 = 4. y = x 2 (2,4)
Ares The motivtion for our definition of integrl ws the problem of finding the re between some curve nd the is for running between two specified vlues. We pproimted the region b union of thin rectngles
Integration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 +
Mth.9 Em Solutions NAME: #.) / #.) / #.) /5 #.) / #5.) / #6.) /5 #7.) / Totl: / Instructions: There re 5 pges nd totl of points on the em. You must show ll necessr work to get credit. You m not use our
to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b.
5.9 Are in rectngulr coordintes If f() on the intervl [; ], then the definite integrl f()d equls to the re of the region ounded the grph of the function = f(), the -is = nd two verticl lines = nd =. =
Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of
Section 7-4 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
Introduction to Integration Part 2: The Definite Integral
Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the
Applications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
Worksheet 24: Optimization
Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I
Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
Let us recall some facts you have learnt in previous grades under the topic Area.
6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us
Sect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its
CONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we
Lesson 10. Parametric Curves
Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find
Pure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
Anti-derivatives/Indefinite Integrals of Basic Functions
Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:
Integration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
Review Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
m, where m = m 1 + m m n.
Lecture 7 : Moments nd Centers of Mss If we hve msses m, m 2,..., m n t points x, x 2,..., x n long the x-xis, the moment of the system round the origin is M 0 = m x + m 2 x 2 + + m n x n. The center of
Area Between Curves: We know that a definite integral
Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region
Approximate Integration: Trapezoid Rule and Simpson s Rule. y 1. e x 2 dx. y b. f sxd dx < o n
Approimte Integrtion: Trpezoid Rule nd Simpson s Rule There re two situtions in which it is impossible to find the ect vlue of definite integrl. The first sitution rises from the fct tht in order to evlute
Operations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
For a solid S for which the cross sections vary, we can approximate the volume using a Riemann sum. A(x i ) x. i=1.
Volumes by Disks nd Wshers Volume of cylinder A cylinder is solid where ll cross sections re the sme. The volume of cylinder is A h where A is the re of cross section nd h is the height of the cylinder.
Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
Uniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A
Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter
Review guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
Answer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your
Brief review of prerequisites for ECON4140/4145
1 ECON4140/4145, August 2010 K.S., A.S. Brief review of prerequisites for ECON4140/4145 References: EMEA: K. Sdsæter nd P. Hmmond: Essentil Mthemtics for Economic Anlsis, 3rd ed., FT Prentice Hll, 2008.
9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
Graphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
Surface Area and Volume
Surfce Are nd Volume Student Book - Series J- Mthletics Instnt Workooks Copyright Surfce re nd volume Student Book - Series J Contents Topics Topic - Surfce re of right prism Topic 2 - Surfce re of right
Pythagoras theorem and trigonometry (2)
HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These
14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the Root-Men-Squre Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
Ae2 Mathematics : Fourier Series
Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl word-for-word with my lectures which will
The Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the single-vrible chin rule extends to n inner
Net Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
Conic Sections. The coefficient of x 2 is positive, so the parabola opens. y up. The vertex is the lowest point on the parabola, or
SECTION. Conic Sections SECTION. Conic Sections OBJECTIVE A To grph prol The conic sections re curves tht cn e constructed from the intersection of plne nd right circulr cone. The prol, which ws introduced
MATLAB: M-files; Numerical Integration Last revised : March, 2003
MATLAB: M-files; Numericl Integrtion Lst revised : Mrch, 00 Introduction to M-files In this tutoril we lern the bsics of working with M-files in MATLAB, so clled becuse they must use.m for their filenme
Binary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
Exponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
Chapter G - Problems
Chpter G - Problems Blinn College - Physics 2426 - Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ10-5 T nd is directed t n
r 2 F ds W = r 1 qe ds = q
Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study
Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
Solutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
Mathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
Numerical Methods of Approximating Definite Integrals
6 C H A P T E R Numericl Methods o Approimting Deinite Integrls 6. APPROXIMATING SUMS: L n, R n, T n, AND M n Introduction Not only cn we dierentite ll the bsic unctions we ve encountered, polynomils,
SUBSTITUTION I.. f(ax + b)
Integrtion SUBSTITUTION I.. f(x + b) Grhm S McDonld nd Silvi C Dll A Tutoril Module for prctising the integrtion of expressions of the form f(x + b) Tble of contents Begin Tutoril c 004 g.s.mcdonld@slford.c.uk
Chapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
Volumes as integrals of cross-sections (Sect. 6.1) Volumes as integrals of cross-sections (Sect. 6.1)
Volumes s integrls of cross-sections (ect. 6.1) Te volume of simple regions in spce Volumes integrting cross-sections: Te generl cse. Certin regions wit oles. Volumes s integrls of cross-sections (ect.
Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review
Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1
Section 5-4 Trigonometric Functions
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
Lesson 12.1 Trigonometric Ratios
Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml
APPLICATION OF INTEGRALS
Chpter 8 APPLICATION OF INTEGRALS 8.1 Overview This chpter dels with specific ppliction of integrls to find the re under simple curves, re etween lines nd rcs of circles, prols nd ellipses, nd finding
Unit 6: Exponents and Radicals
Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -
Chapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
The Velocity Factor of an Insulated Two-Wire Transmission Line
The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
10.6 Applications of Quadratic Equations
10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,
An Off-Center Coaxial Cable
1 Problem An Off-Center Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor
Using Definite Integrals
Chpter 6 Using Definite Integrls 6. Using Definite Integrls to Find Are nd Length Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: How
Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
Two special Right-triangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of ) - c - (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Right-tringles. The
y 20 x 1 Solution: a. The x-coordinates of the intersection points are solutions to the equation f(x) = g(x).
Mth 8, Exm, Fll 4 Problem Solution. Consider the functions fx) = x 6x nd gx) = 6x 4.. Find the x-coordinte of the intersection points of these two grphs. b. Compute the re of the region bounded by the
11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
Algorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
EQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
Math 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction
PHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
Square Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
Or more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
Lecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
Section A-4 Rational Expressions: Basic Operations
A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the
Sequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the Euler-Lgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
Unit 4. Applications of integration
SOLUTIONS TO 8. EXERCISES Unit 4. Applictions of integrtion 4A. Ares between curves. / 4A- ) (3x x )dx = (3/)x x (/3)x 3 / = /4 b) x 3 = x = x = ± or x =. There re two enclosed pieces ( < x < nd < x
CHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
Math 22B Solutions Homework 1 Spring 2008
Mth 22B Solutions Homework 1 Spring 2008 Section 1.1 22. A sphericl rindrop evportes t rte proportionl to its surfce re. Write differentil eqution for the volume of the rindrop s function of time. Solution
PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
Notes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams
Notes for Thurs 8 Sept Clculus II Fll 00 New York University Instructor: Tyler Neylon Scribe: Kelsey Willims 8. Integrtion by Prts This section is primrily bout the formul u dv = uv v ( ) which is essentilly
The Fundamental Theorem of Calculus
Section 5.4 Te Funmentl Teorem of Clculus Kiryl Tsiscnk Te Funmentl Teorem of Clculus EXAMPLE: If f is function wose grp is sown below n g() = f(t)t, fin te vlues of g(), g(), g(), g(3), g(4), n g(5).
Diffraction and Interference of Light
rev 12/2016 Diffrction nd Interference of Light Equipment Qty Items Prt Number 1 Light Sensor CI-6504 1 Rotry Motion Sensor CI-6538 1 Single Slit Set OS-8523 1 Multiple Slit Set OS-8523 1 Liner Trnsltor