# 7.6 The Use of Definite Integrals in Physics and Engineering

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

### Math 1132 Worksheet 6.4 Name: Discussion Section: 6.4 Work

Mth 1132 Worksheet 6.4 Nme: Discussion Section: 6.4 Work Force formul for springs. By Hooke s Lw, the force required to mintin spring stretched x units beyond its nturl length is f(x) = kx where k is positive

### 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

### a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

### Math 120 Answers for Homework 13

Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing

### We divide the interval [a, b] into subintervals of equal length x = b a n

Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

### APPLICATIONS OF THE DEFINITE INTEGRAL

APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through

### 13.4 Work done by Constant Forces

13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

### [ ( ) ( )] 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.

### Sample Problems for the Final of Math 121, Fall, 2005

Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.

### Definite integral. Mathematics FRDIS MENDELU

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

### Centre of Mass, Moments, Torque

Centre of ss, oments, Torque Centre of ss If you support body t its center of mss (in uniform grvittionl field) it blnces perfectly. Tht s the definition of the center of mss of the body. If the body consists

### Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

### JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 8 (First moments of a volume) A.J.Hobson

JUST THE MATHS UNIT NUMBER 3.8 INTEGRATIN APPLICATINS 8 (First moments of volume) b A.J.Hobson 3.8. Introduction 3.8. First moment of volume of revolution bout plne through the origin, perpendiculr to

### INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

### Final Exam - Review MATH Spring 2017

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

### (6.5) Length and area in polar coordinates

86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +

### Math 116 Final Exam April 26, 2013

Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems

### 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

### The Fundamental Theorem of Calculus

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

### 1 The Riemann Integral

The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

### MAT187H1F Lec0101 Burbulla

Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is

### 6.5 Numerical Approximations of Definite Integrals

Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,

### 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

### 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

### 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 Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F

### Section 6: Area, Volume, and Average Value

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

### JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson

JUST THE MATHS UNIT NUMBE 13.1 INTEGATION APPLICATIONS 1 (Second moments of n re (B)) b A.J.Hobson 13.1.1 The prllel xis theorem 13.1. The perpendiculr xis theorem 13.1.3 The rdius of grtion of n re 13.1.4

### Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

### 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

### 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...

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### 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

### 7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

### PREVIOUS EAMCET QUESTIONS

CENTRE OF MASS PREVIOUS EAMCET QUESTIONS ENGINEERING Two prticles A nd B initilly t rest, move towrds ech other, under mutul force of ttrction At n instnce when the speed of A is v nd speed of B is v,

### STATICS. Vector Mechanics for Engineers: Statics VECTOR MECHANICS FOR ENGINEERS: Centroids and Centers of Gravity.

5 Distributed CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinnd P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Wlt Oler Texs Tech Universit Forces: Centroids nd Centers of Grvit Contents Introduction

### 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

### JUST THE MATHS SLIDES NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson

JUST THE MATHS SLIDES NUMBER 13.12 INTEGRATION APPLICATIONS 12 (Second moments of n re (B)) b A.J.Hobson 13.12.1 The prllel xis theorem 13.12.2 The perpendiculr xis theorem 13.12.3 The rdius of grtion

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

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

### Chapters 4 & 5 Integrals & Applications

Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

### Section Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?

Section 5. - Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles

### 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:

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### 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:

### Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

### 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

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

### Geometric and Mechanical Applications of Integrals

5 Geometric nd Mechnicl Applictions of Integrls 5.1 Computing Are 5.1.1 Using Crtesin Coordintes Suppose curve is given by n eqution y = f(x), x b, where f : [, b] R is continuous function such tht f(x)

### Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.

Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite

### 5 Applications of Definite Integrals

5 Applictions of Definite Integrls The previous chpter introduced the concepts of definite integrl s n re nd s limit of Riemnn sums, demonstrted some of the properties of integrls, introduced some methods

### 10 Vector Integral Calculus

Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

### 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:

### 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

### 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

### PhysicsAndMathsTutor.com

1. A uniform circulr disc hs mss m, centre O nd rdius. It is free to rotte bout fixed smooth horizontl xis L which lies in the sme plne s the disc nd which is tngentil to the disc t the point A. The disc

### 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

### F is on a moving charged particle. F = 0, if B v. (sin " = 0)

F is on moving chrged prticle. Chpter 29 Mgnetic Fields Ech mgnet hs two poles, north pole nd south pole, regrdless the size nd shpe of the mgnet. Like poles repel ech other, unlike poles ttrct ech other.

### Integrals - Motivation

Integrls - Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is non-liner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

### KINEMATICS OF RIGID BODIES

KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016

Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric

### CHAPTER 5 Newton s Laws of Motion

CHAPTER 5 Newton s Lws of Motion We ve been lerning kinetics; describing otion without understnding wht the cuse of the otion ws. Now we re going to lern dynics!! Nno otor 103 PHYS - 1 Isc Newton (1642-1727)

### 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

### 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,

### AB Calculus Review Sheet

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

### MA 124 January 18, Derivatives are. Integrals are.

MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,

### ME 141. Lecture 10: Kinetics of particles: Newton s 2 nd Law

ME 141 Engineering Mechnics Lecture 10: Kinetics of prticles: Newton s nd Lw Ahmd Shhedi Shkil Lecturer, Dept. of Mechnicl Engg, BUET E-mil: sshkil@me.buet.c.bd, shkil6791@gmil.com Website: techer.buet.c.bd/sshkil

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

### 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

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

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

### JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson

JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson 13.6.1 Introduction 13.6. First moment of n rc bout the y-xis 13.6.3 First moment of n rc bout the x-xis

### Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

### 8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS

8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS 8 Arc Length nd Surfce Are Preliminr Questions Which integrl represents the length of the curve cos between nd π? π π + cos d, + sin d Let

### ( ) as a fraction. Determine location of the highest

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

### Simple Harmonic Motion I Sem

Simple Hrmonic Motion I Sem Sllus: Differentil eqution of liner SHM. Energ of prticle, potentil energ nd kinetic energ (derivtion), Composition of two rectngulr SHM s hving sme periods, Lissjous figures.

### 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

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

### Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...

Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.

### CHAPTER 4 MULTIPLE INTEGRALS

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

### The momentum of a body of constant mass m moving with velocity u is, by definition, equal to the product of mass and velocity, that is

Newtons Lws 1 Newton s Lws There re three lws which ber Newton s nme nd they re the fundmentls lws upon which the study of dynmics is bsed. The lws re set of sttements tht we believe to be true in most

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### 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.

### Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

### Applications of Bernoulli s theorem. Lecture - 7

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

### Problem Set 3 Solutions

Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,

### 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,

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

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

### Distributed Forces: Centroids and Centers of Gravity

Distriuted Forces: Centroids nd Centers of Grvit Introduction Center of Grvit of D Bod Centroids nd First Moments of Ares nd Lines Centroids of Common Shpes of Ares Centroids of Common Shpes of Lines Composite

### ROB EBY Blinn College Mathematics Department

ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob Eby-Fll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous

### n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

### Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc

Physics 170 Summry of Results from Lecture Kinemticl Vribles The position vector ~r(t) cn be resolved into its Crtesin components: ~r(t) =x(t)^i + y(t)^j + z(t)^k. Rtes of Chnge Velocity ~v(t) = d~r(t)=