# 7.6 The Use of Definite Integrals in Physics and Engineering

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems such s problems concerned with computing re, volume, surfce re. nd rc length. In this section clculus is used to solve problems tht rise from Physics nd engineering. The Concept of Work The work done by constnt force, F, in moving n object distnce, d, is equl to the product of the force nd the distnce moved. Tht is, W = F d. The SI (interntionl) unit of work is the joule (J), which is the work done by force of one Newton (N) pushing body long one meter (m). Thus, 1 joule = 1 N-m. In the British system, unit work is the foot-pound. Since 1N =.489 lb(1 lb = 4.45 N) nd 1m = ft (1 ft =.35 m), we hve 1J = ft lb(1 ft lb = 1.36 J). Now, in most cses the pplied force is not constnt, but vries over the stright line of motion. For exmple, suppose tht the force, F (x), cting on prticle s it moves long the stright line from to b vries continuously. In order to find the totl work done by the force we divide the intervl [, b] into n smll equl subintervls [x i 1, x i ], ech of length x, so tht the chnge in F is smll long ech subintervl, i.e., pproximtely constnt. Then the work done by the force in moving the body from x i 1 to x i is pproximtely: W i F (x i ) x where x i 1 x i x i. Hence, the totl work is W F (x i ) x. As n the Riemnn sum t the right converges to the following integrl: W = F (x)dx. Remrk In physics, the kinetic energy of n object is the energy which it possesses due to its motion. It is defined s the work needed to ccelerte body 1

2 of given mss from rest to its stted velocity. Hving gined this energy during its ccelertion, the body mintins this kinetic energy unless its speed chnges. The sme mount of work is done by the body in decelerting from its current speed to stte of rest. Exmple Consider spring on the x xis so tht its right end is t when the spring is t its rest position. According to Hooke s Lw, the force needed to stretch the spring from to x is proportionl to x, i.e., F (x) = kx where k is clled the spring constnt. See Figure Figure Find the work done in stretching the spring length of. The work needed to stretch the spring from to is given by the integrl W = kxd k Exmple 7.6. (Work Done Filling (or Emptying) Tnk) A tnk in the shpe of right circulr cone of height 1 m nd rdius 4 m is inserted into the ground with its vertex pointing down nd its top t ground level. If the tnk is filled with wter (density ρ = 1kg/m 3 ) to depth of 8 m, how much work is performed in pumping ll the wter in the tnk to ground level? Wht chnges if the wter is pumped to height of m bove ground level? Set up coordinte system s shown in Figure 7.6..

3 Figure 7.6. Consider lyer of distnce x i from the bse of the cone nd with thickness x. The volume of such circulr lyer is Using similr tringles we find tht V i πr i x. r i 4 = 1 x i 1 nd consequently r i = 5 (1 x i ). Thus, V i 4π 5 (1 x i ) x. Hence its mss is m i = 1 4π 5 (1 x i ) 16π(1 x i ) x. The force required to rise this lyer is f i = m i g = 9.8[16π(1 x i ) x] = 1568π(1 x i ) x. The work done to rise it to the top of the tnk is W i 1568π(1 x i ) x i x. Adding the works done to rise these slices we obtin the totl work done to empty the tnk: W = π(1 x) xd J. 3

4 Now if the wter is pumped to height of m bove ground level then W = π(1 x) (x + )d J Force nd Pressure Pressure is the force per unit re cting on n object. The pressure is exerted eqully in ll directions nd it increses in depth. Consider thin horizontl plte of re A squre meters submerged in liquid t given depth d below the surfce. The volume of the liquid directly bove the plte is V = Ad nd its mss is m = ρad where ρ is the density of the liquid. Thus, the hydrosttic force exerted by the liquid on the plte is F = mg = ρgad where g is the ccelertion due to grvity. Hence, the pressure P on the plte is defined by P = F A = ρgd. The SI unit for pressure is clled Pscl. Thus, 1 P = 1 Newton per squre meter. Exmple Consider dm for storing wter s shown in Figure Set up nd clculte definite integrl giving the totl hydrosttic force on the dm if wter level is 4 m from the top of the dm. The density of wter is ρ = 1 kg/m 3. Figure We divide the dm into horizontl strips in which the pressure is lmost constnt. Let s find the re of the i th strip which is pproximtely rectngle 4

5 of height x nd width w i = (15 + ). From similr tringles, we hve 1 = 16 x i = = 8.5x i. Thus, A i (3.5x i ) (46 x i ) x. The pressure exerted on this strip of the dm is given by P i 1gx i. The hydrosttic force cting on this strip is F i = P i A i 1gx i (46 x i ) x. Thus, the totl force is F = 16 1gx(46 x)d N Moments nd Center of Mss In this section we wnt to find point on which thin plte of ny given shpe blnces horizontlly s in Figure Figure The center of mss is the so-clled blncing point of n object (or system.) For exmple, when two children re sitting on seesw, the point t which the seesw blnces, i.e. becomes horizontl is the center of mss of the seesw. 5

6 Discrete Point Msses: One Dimensionl Cse Consider gin the exmple of two children of mss m 1 nd m sitting on ech side of seesw. It cn be shown experimentlly tht the center of mss is point P on the seesw such tht m 1 d 1 = m d where d 1 nd d re the distnces from m 1 nd m to P respectively. See Figure In order to generlize this concept, we introduce n x xis with points m 1 nd m locted t points with coordintes x 1 nd x. Figure Since P is the blncing point, we must hve Solving for x we find m 1 (x x 1 ) = m (x x). m 1x 1 + m x. m 1 + m The product m i x i is clled the moment of m i bout the origin. The bove result cn be extended to system with mny points s follows: The center of mss of system of n point-msses m 1, m,, m n locted t x 1, x,, x n long the x xis is given by the formul m i x i The sum M = origin. m i m i x i is clled the moment of the system bout the 6

7 Exmple Point msses m i re locted on the x xis s shown in Figure Find the moment M of the system bout the origin nd the center of mss x. Figure The moment of the system bout the origin is M = 1( 3) + 15() + (8) = 154. The center of mss is = Discrete System: Two dimensionl cse The concept of center of mss cn be pplied to two dimensionl objects s well. The determintion of the center of mss in two dimensions is done in similr mnner. If mss m is locted t point (x, y) then we define the moment of m bout the x xis to be the product my nd the moment of m bout the y xis to be the product mx. Let (x, y) be the center of mss. The procedure of finding formuls for x nd y is the sme s the one dimensionl cse. Add up the msses times their x loctions then divide by totl mss to get x. Next, dd up the msses times their y loctions then divide by totl mss to get y. Hence the two formuls: x i m i y i m i where M nd M y = My m = m i nd y = Mx m = y i m i is the moment of the system bout the x xis x i m i is the moment of the system bout the y xis. Since m M y nd my = M x, the center of mss (x, y) is the point where single prticle of mss m would hve the sme moments s the system. 7 m i

8 Exmple Point msses m i re locted t the points P i. Find the moment M x nd M y nd the center of mss of the system: We hve m i P i 4 (, 3) ( 3, 1) 4 (3, 5) M 4( 3) + (1) + 4(5) = 1 M y =4() + ( 3) + 4(3) = 14 m = = = 1.4 y = 1 1 = 1 Continuous System:One Dimensionl Cse Next we consider continuous system. Suppose tht we hve n object lying on the x xis between nd b. At point x, suppose tht the object hs mss density (mss per unit length) of δ(x). To clculte the center of mss, we divide the object into n pieces, ech of length x. On ech piece, the density is nerly constnt, so the mss of the i th piece is m i δ(x i ) x. The center of mss is then m i x i Letting n we obtin m i x i δ(x i ) x. δ(x i ) x xδ(x)dx δ(x)dx. 8

9 Exmple Find the center of mss of -meter rod lying on the x xis with its left end t the origin if its density is δ(x) = 15x kg/m. The totl mss is The center of mss is M = 15x d 5x 3 = 4 kg. 15x3 dx = [ x 4 4 ] = 1.5 m. Two Dimensionl System: Continuous cse In the continuous cse, we consider thin plte tht occupies region in the plne s shown in Figure We ssume the plte hs uniform density ρ. Figure Divide the intervl [, b] into n subintervls with endpoints x i = + i x nd length b n. Let x i = x i = x i 1+x i. Then the center of mss of the rectngle R i is C i (x i, 1 f(x i)). The mss of this rectngle is m i = ρf(x i ) x. Thus, nd we define M y = lim n M y (R i ) = ρf(x i ) xx i ρf(x i )x i ρxf(x)dx. 9

10 Likewise, Thus, nd M lim n ρ 1 [f(x i)] ρxf(x)dx ρ f(x)d xf(x)dx b f(x)dx y = ρ 1 [f(x)] dx ρ f(x)d 1 ρ[f(x)] dx. 1 [f(x)] dx f(x)dx. Exmple Find the center of mss of semicirculr plte of rdius r. Figure Due to symmetry, the center of mss must lie on the y xis so tht. Now, y = r r = πr = 4r 3π 1 [ r x ] dx πr [r x x3 3 Exmple Find the center of mss of the region bounded by the line y = x nd the ] r r 1

11 prbol y = x. Figure We hve A = 6 y = (x x )d 1 6 x(x x )d 1 1 (x x 4 )d 5 Remrk 7.6. The center of mss of body need not be within the body itself; the center of mss of ring or hollow cylinder of uniform density is locted in the enclosed spce, not in the object itself. 11

### l 2 p2 n 4n 2, the total surface area of the

Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

### Math 0230 Calculus 2 Lectures

Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

### 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

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### [ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

### Week 10: Riemann integral and its properties

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

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

Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

### Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

### APPM 1360 Exam 2 Spring 2016

APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

### FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...

### Math 113 Exam 1-Review

Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

### 38 Riemann sums and existence of the definite integral.

38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

### 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

### Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

### 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:

### Calculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties

Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwth-chen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:

### Big idea in Calculus: approximation

Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

### Not for reproduction

AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type

### x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

### Shape and measurement

C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

### Distance And Velocity

Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

### ragsdale (zdr82) HW2 ditmire (58335) 1

rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

### Conducting Ellipsoid and Circular Disk

1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,

### Math 113 Exam 2 Practice

Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This

### APPLICATIONS OF INTEGRATION

6 APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6.4 Work In this section, we will learn about: Applying integration to calculate the amount of work done in performing a certain physical task.

### E S dition event Vector Mechanics for Engineers: Dynamics h Due, next Wednesday, 07/19/2006! 1-30

Vector Mechnics for Engineers: Dynmics nnouncement Reminders Wednesdy s clss will strt t 1:00PM. Summry of the chpter 11 ws posted on website nd ws sent you by emil. For the students, who needs hrdcopy,

### 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

### Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:

Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You

### Math 554 Integration

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

### Mathematics of Motion II Projectiles

Chmp+ Fll 2001 Dn Stump 1 Mthemtics of Motion II Projectiles Tble of vribles t time v velocity, v 0 initil velocity ccelertion D distnce x position coordinte, x 0 initil position x horizontl coordinte

### x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

### Section 14.3 Arc Length and Curvature

Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

### 5.2 Volumes: Disks and Washers

4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

### 200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

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

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

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

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

### PHYSICS 211 MIDTERM I 21 April 2004

PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### Lecture 1: Electrostatic Fields

Lecture 1: Electrosttic Fields Instructor: Dr. Vhid Nyyeri Contct: nyyeri@iust.c.ir Clss web site: http://webpges.iust.c. ir/nyyeri/courses/bee 1.1. Coulomb s Lw Something known from the ncient time (here

### Physics 2135 Exam 1 February 14, 2017

Exm Totl / 200 Physics 215 Exm 1 Ferury 14, 2017 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the est or most nerly correct nswer. 1. Two chrges 1 nd 2 re seprted

### Dynamics: Newton s Laws of Motion

Lecture 7 Chpter 4 Physics I 09.25.2013 Dynmics: Newton s Lws of Motion Solving Problems using Newton s lws Course website: http://fculty.uml.edu/andriy_dnylov/teching/physicsi Lecture Cpture: http://echo360.uml.edu/dnylov2013/physics1fll.html

### Riemann Integrals and the Fundamental Theorem of Calculus

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

### Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

### The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

### STATICS VECTOR MECHANICS FOR ENGINEERS: and Centers of Gravity. Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.

007 The McGrw-Hill Compnies, Inc. All rights reserved. Eighth E CHAPTER 5 Distriuted VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinnd P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Wlt Oler Tes Tech

### Phys 4321 Final Exam December 14, 2009

Phys 4321 Finl Exm December 14, 2009 You my NOT use the text book or notes to complete this exm. You nd my not receive ny id from nyone other tht the instructor. You will hve 3 hours to finish. DO YOUR

### 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,

### UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

### Math 100 Review Sheet

Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

### I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

### Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

### 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

### = 40 N. Q = 60 O m s,k

Multiple Choice ( 6 Points Ech ): F pp = 40 N 20 kg Q = 60 O m s,k = 0 1. A 20 kg box is pulled long frictionless floor with n pplied force of 40 N. The pplied force mkes n ngle of 60 degrees with the

### MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

### Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014

Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method

### Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

### (0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35

7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know

### Contour and surface integrals

Contour nd surfce integrls Contour integrls of the sclr type These re integrls of the type I = f@rd l C where C is contour nd l is n infinitesiml element of the contour length. If the contour cn be described

Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

### Flow in porous media

Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ

### 20 MATHEMATICS POLYNOMIALS

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

### Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

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

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

### MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008

MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul

### Chapter 9 Definite Integrals

Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

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

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

### Physics Honors. Final Exam Review Free Response Problems

Physics Honors inl Exm Review ree Response Problems m t m h 1. A 40 kg mss is pulled cross frictionless tble by string which goes over the pulley nd is connected to 20 kg mss.. Drw free body digrm, indicting

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

### Resistors. Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ( )

10/25/2005 Resistors.doc 1/7 Resistors Consider uniform cylinder of mteril with mediocre to poor to r. pthetic conductivity ( ) ˆ This cylinder is centered on the -xis, nd hs length. The surfce re of the

### III. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9

III Lecture on Numericl Integrtion File fclib/dttb/lecture-notes/numerical-inter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

### This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

### Applications of Integration

9 Chpter 9 Applictions of Integrtion 9 Applictions of Integrtion Ö ØÛ Ò ÙÖÚ º½ We hve seen how integrtion cn be used to find n re between curve nd the x-xis With very little chnge we cn find some res between

### f (x) dx = f(b) f(a) f (x i ) x i i=1

1 Cse Study: Flood Wtch c 00 Donld Kreider nd Dwight L Animtion: Flood Wtch To get you going on the Cse Study! In this section, we hve lerned tht if we re given the continuous derivtive f of function on

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### Quantum Physics I (8.04) Spring 2016 Assignment 8

Quntum Physics I (8.04) Spring 206 Assignment 8 MIT Physics Deprtment Due Fridy, April 22, 206 April 3, 206 2:00 noon Problem Set 8 Reding: Griffiths, pges 73-76, 8-82 (on scttering sttes). Ohnin, Chpter

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### Lecture 3. Limits of Functions and Continuity

Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

### DIRECT CURRENT CIRCUITS

DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

### MATH 115: Review for Chapter 7

MATH 5: Review for Chpter 7 Cn you stte the generl form equtions for the circle, prbol, ellipse, nd hyperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol

### CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

### Some Methods in the Calculus of Variations

CHAPTER 6 Some Methods in the Clculus of Vritions 6-. If we use the vried function ( α, ) α sin( ) + () Then d α cos ( ) () d Thus, the totl length of the pth is d S + d d α cos ( ) + α cos ( ) d Setting

### A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

### Correct answer: 0 m/s 2. Explanation: 8 N

Version 001 HW#3 - orces rts (00223) 1 his print-out should hve 15 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Angled orce on Block 01 001

### Interpreting Integrals and the Fundamental Theorem

Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

### 5.6 Work. Common Units Force Distance Work newton (N) meter (m) joule (J) pound (lb) foot (ft) Conversion Factors

5.6 Work Page 1 of 7 Definition of Work (Constant Force) If a constant force of magnitude is applied in the direction of motion of an object, and if that object moves a distance, then we define the work

### Discrete Least-squares Approximations

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

### We know that if f is a continuous nonnegative function on the interval [a, b], then b

1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

### Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

### MA Handout 2: Notation and Background Concepts from Analysis

MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

### Orthogonal Polynomials and Least-Squares Approximations to Functions

Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

### CHAPTER 1 CENTRES OF MASS

1.1 Introduction, nd some definitions. 1 CHAPTER 1 CENTRES OF MASS This chpter dels with the clcultion of the positions of the centres of mss of vrious odies. We strt with rief eplntion of the mening of