Double Integrals over General Regions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Double Integrals over General Regions"

Transcription

1 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 verticll: Slicing verticll mounts to slicing the intervl [, ] on the x-xis, so our outer integrl will be something dx. To figure out the inner integrl, we look t generl slice. emember tht, on single slice, x is (roughl constnt, nd we wnt to describe wht does. The bottom of ech slice is on the line, nd the top is on the line x, so the inner integrl hs endpoints of integrtion nd x. Therefore, our iterted integrl is x x d dx x x ( x x x dx x dx Slicing horizontll: Slicing horizontll mounts to slicing the intervl [, ] on the -xis, so our outer integrl will be something d. To figure out the inner integrl, we look t generl slice. The left end of ech slice is on the line x, nd the right end is on the line x. Since we re describing ( emember tht this is stremlined version of the rel process. ell, to get iemnn sum pproximtion, we chop the region into lots of smll rectngles, ech of width x nd height. The re of ech piece is then A x. We hve one product f(x, x per little rectngle, nd we need to dd these ll up to get iemnn sum. (See # of the worksheet Double Integrls for more detils. When converting to n iterted integrl, we re rell deciding whether we wnt to dd up in rows or columns first. If we dd up in rows, we visulize dding up in horizontl slice first nd getting one sum per horizontl slice (then we dd up ll of those sums, one per slice. Similrl, if we dd up in columns, we visulize dding up in verticl slice first nd then dding up ll those sums, one per verticl slice. So, when we s slice horizontll, we rell men we re going to dd up in rows first.

2 horizontl slice, we wnt to describe how x vries, so x goes from to. Thus, the iterted integrl is / x dx d, which is of course lso equl to.. Let be the region bounded b x nd. Write the double integrl f(x, dx d s n iterted integrl in both possible orders. Solution. Agin, we think of slicing either verticll or horizontll. x x Slicing verticll: Slicing verticll mounts to slicing the intervl [, ] on the x-xis, so the outer integrl will be something dx. To write the inner integrl, we wnt to describe wht does within single slice (thinking of x s being constnt. The bottom of ech slice lies on x, nd the top lies on, so the iterted integrl is Slicing horizontll: x f(x, d dx. Slicing horizontll mounts to slicing the intervl [, ] on the -xis, so the outer integrl will be something d. The left side of ech slice lies on x, nd the right side of ech slice lso lies on x. emember, though, tht we re tring to describe how x vries in slice (nd we think of s being constnt, so x goes from the left hlf of x, where x, to the right hlf, where x. Thus, the iterted integrl is f(x, dx d.. Let be the trpezoid with vertices (,, (,, (,, nd (,. Write the double integrl f(x, dx d s n iterted integrl. Solution. Let s compre slicing verticll with slicing horizontll: x x Notice tht, if we slice verticll, there re two tpes of slices. The slices to the left of x go from to, wheres the slices to the right go from to the digonl side of the trpezoid.

3 In contrst, if we slice horizontll, ll of the slices hve the sme description: the go from x to the digonl side. This seems simpler, so let s go with this method. When we slice horizontll, we re slicing the intervl [, ] on the -xis, so our outer integrl will be something d. Ech slice goes from x to the digonl side. The digonl side is x (we know it s line contining the points (, nd (,. We wnt to describe how x vries in ech slice, so x goes from to. So, the iterted integrl is. Evlute the double integrl f(x, dx d. ( + dx d where is the region in the first qudrnt bounded b x,, nd x. (To decide the order of integrtion, first think bout whether it s esier to integrte the integrnd with respect to x or with respect to. Solution. The integrnd is much esier to integrte with respect to x thn with respect to. Therefore, we should tr to rewrite the double integrl s n iterted integrl where the inner integrl is with respect to x. This mens our outer integrl will be with respect to, which corresponds in our strteg to slicing the region horizontll. x This mounts to slicing the intervl [, ] on the -xis, so the outer integrl will be something d. Ech slice hs its left end on x nd its right end on x. We wnt to describe how x vries within slice, so we rewrite x s x. This gives the iterted integrl + dx d ( x + + d x x We cn evlute this integrl using substitution: if we let u +, then du d, nd we cn rewrite the integrl s u u du 9 u/ u ( / 9 d ( If ou used the other order of integrtion, ou should hve sum of iterted integrls x f(x, d dx. f(x, d dx +

4 5. In ech prt, ou re given n iterted integrl. Sketch the region of integrtion, nd then chnge the order of integrtion. ( x f(x, d dx. Solution. Let s just think of our strteg in reverse. The fct tht the outer integrl is something dx tells us tht we re slicing the intervl [, ] on the x-xis, so we re mking verticl slices from x to x. The inner integrl tells us tht the bottom of ech slice is on, nd the top of ech slice is on x. So, the region of integrtion (with verticl slices looks like the picture on the left: x x x x (b To chnge the order of integrtion, we wnt to insted use horizontl slices (the picture on the right. Now, we re slicing the intervl [, ] on the -xis, so the outer integrl is something d. Ech slice hs its left edge on x (or x, since we rell wnt to describe x in terms of nd its right edge on x, so we cn rewrite the iterted integrl s f(x, dx d. Solution. The fct tht the outer integrl is f(x, dx d. something d tells us tht we re slicing the intervl [, ] on the -xis, so we re mking horizontl slices from to. The inner integrl tells us tht the left side of ech slice is on x nd the right side is on x (or x. So, the region of integrtion looks like this: x x x x To chnge the order of integrtion, we use verticl slices. Now, we re slicing the intervl [, ] on the x-xis. The bottom of ech slice is on x, nd the top of ech slice is on, so we cn rewrite the integrl s x f(x, d dx.

5 (c f(x, dx d. Solution. The fct tht the outer integrl is something d tells us tht we re slicing the intervl [, ] on the -xis, so we re mking horizontl slices from to. The inner integrl tells us tht the left side of ech slice is on x nd the right side of ech slice is on x. x describes the left hlf of the circle x +, nd x describes the right hlf, so the region of integrtion looks like this: x x To chnge the order of integrtion, we use verticl slices. Now, we re slicing the intervl [, ] on the x-xis, so the outer integrl is something dx. Ech slice hs its bottom edge on nd its top edge on the top hlf of the circle x + (or x, so we cn rewrite the iterted integrl s x f(x, d dx. 6. Let be constnt between nd. Let be the region bounded b x + nd. Write the double integrl f(x, dx d s n iterted integrl in both possible orders. Solution. The curves x + nd intersect where x, so x ±. So, the region looks like this:,, x To write the double integrl s n iterted integrl, we think of slicing either verticll or horizontll. 5

6 ,,,, x x Slicing verticll: Slicing verticll corresponds to slicing the intervl [, ] on the x-xis, so the outer integrl will be edge on, so the iterted integrl is something dx. Ech slice hs its bottom edge on x + nd its top constnt, so it s fine to hve it in the outer integrl. Slicing horizontll: x + f(x, d dx. emember tht is Slicing horizontll corresponds to slicing the intervl [, ] on the -xis, so the outer integrl will be something d. Ech slice hs its left edge on x + (so x nd its right edge on x + (so x. Thus, the iterted integrl is 7. Evlute the iterted integrl x x cos ( d dx. f(x, dx d. Solution. We don t know how to integrte the integrnd with respect to, but we cn integrte it with respect to x. This suggests tht we should chnge the order of integrtion, s in??. First, let s figure out wht the region looks like. The fct tht the outer integrl is something dx tells us tht we re slicing the intervl [, ] on the x-xis, so we re mking verticl slices from x to x. The inner integrl tells us tht the bottom of ech slice is on x (the bottom hlf of the circle x + nd the top of ech slice is on. So, the region of integrtion looks like this: x x 6

7 To chnge the order of integrtion, we switch to using horizontl slices. Now, we re slicing the intervl [, ] on the -xis, so our outer integrl will be something d. Ech slice hs its left edge on x nd its right edge on the right hlf of the circle x + (so x. Therefore, we cn rewrite the given integrl s x cos ( dx d x cos ( d ( cos ( We cn use substitution to evlute this integrl: let u ; then, du ( d, so the integrl becomes cos u du sin u / sin u u / ( d 8. A flt plte is in the shpe of the region in the first qudrnt bounded b x,, ln x nd. If the densit of the plte t point (x, is xe grms per cm, find the mss of the plte. (Suppose the x- nd -xes re mrked in cm. Solution. As we lerned in #(b of the worksheet Double Integrls, we cn find the mss of the plte b tking the double integrl of the densit, where the region of integrtion is the plte. In this cse, the integrnd xe is es to integrte with respect to x nd with respect to, so we will pick n order of integrtion bsed on the shpe of the region. We cn either slice horizontll or verticll: x x As in??, this region is simpler to describe using horizontl slices: with verticl slices, there re two tpes of slices, but with horizontl slices, there is onl one. If we use horizontl slices, we re slicing the intervl [, ] on the -xis. Ech slice goes from x to ln x (or x e, so the iterted integrl is e ( xe xe dx d x e d e d 6 e 6 ( e 6 x 7

8 9. Let U be the solid bove z, below z, nd between the surfces x sin nd x sin +. Find the volume of U. Solution. The picture on the left shows the four surfces z, z, x sin, nd x sin +. The picture on the right shows just the solid U. z z x x This solid cn be described s the solid under z over the region, where is where the solid meets the x-plne. So, its volume will just be ( dx d. To clculte this double integrl, we need to describe nd convert the double integrl to n iterted integrl. The surfce z intersects the x-plne z where, or ±, so nd re boundx dries of the region. The other two re x sin nd x sin +. So, looks like this: x sin x x sin It s esier to slice this region horizontll: x sin x x sin This mounts to slicing the intervl [, ] on the -xis, so the outer integrl will be something d. 8

9 The left side of ech slice is on x sin, nd the right side is on x sin +, so we cn rewrite the double integrl s n iterted integrl sin + sin ( dx d 8 [ x( ( d xsin + xsin ] d 6 9

Area Between Curves: We know that a definite integral

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

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION 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.

More information

1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 +

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

More information

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)

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

More information

m, where m = m 1 + m m n.

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

More information

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. 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

More information

6.2 Volumes of Revolution: The Disk Method

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

More information

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

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

More information

y 20 x 1 Solution: a. The x-coordinates of the intersection points are solutions to the equation f(x) = g(x).

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

More information

Lesson 10. Parametric Curves

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

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

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:

More information

Arc Length. P i 1 P i (1) L = lim. i=1

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

More information

Integration by Substitution

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

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

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

More information

Volumes of solids of revolution

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

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

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

More information

Applications to Physics and Engineering

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

More information

Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

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

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Review Problems for the Final of Math 121, Fall 2014

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

More information

Sequences and Series

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

More information

MATLAB: M-files; Numerical Integration Last revised : March, 2003

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

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

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.

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.

More information

Graphs on Logarithmic and Semilogarithmic Paper

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

More information

Math 1105: Calculus II (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 5

Math 1105: Calculus II (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 5 Mth 5: Clculus II Mth/Sci mjos) MWF m / pm, Cmpion 35 Witten homewok 5 6.6, p. 458 3,33), 6.7, p. 467 8,3), 6.875), 7.58,6,6), 7.44,48) Fo pctice not to tun in): 6.6, p. 458,8,,3,4), 6.7, p. 467 4,6,8),

More information

Review guide for the final exam in Math 233

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

More information

Sect 8.3 Triangles and Hexagons

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

More information

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 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

More information

Net Change and Displacement

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

More information

Solutions to Section 1

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

More information

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes

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

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

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

More information

Worksheet 24: Optimization

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

More information

1 Numerical Solution to Quadratic Equations

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

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

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

More information

Uniform convergence and its consequences

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,

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

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.

More information

CONIC SECTIONS. Chapter 11

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

More information

An Off-Center Coaxial Cable

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

More information

Answer, Key Homework 8 David McIntyre 1

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

More information

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.

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 =. =

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Using Definite Integrals

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

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

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

More information

Answer, Key Homework 4 David McIntyre Mar 25,

Answer, Key Homework 4 David McIntyre Mar 25, Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his print-out should hve 18 questions. Multiple-choice questions my continue on the next column or pe find ll choices before mkin your selection.

More information

Math 135 Circles and Completing the Square Examples

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

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

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.

More information

Chapter G - Problems

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

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

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 +

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Introduction to Integration Part 2: The Definite Integral

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

More information

Operations with Polynomials

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

More information

Pythagoras theorem and trigonometry (2)

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

More information

Complementary Coffee Cups

Complementary Coffee Cups Complementry Coffee Cups Thoms Bnchoff Tom Bnchoff (Thoms Bnchoff@brown.edu) received his B.A. from Notre Dme nd his Ph.D. from the University of Cliforni, Berkeley. He hs been teching t Brown University

More information

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.

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

More information

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

Integration. 148 Chapter 7 Integration

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

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

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

More information

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time.

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time. AP QUIZ # GRAPHING MOTION ) POSITION TIME GRAPHS DISPLAEMENT Ech grph below shows the position of n object s function of time. A, B,, D, Rnk these grphs on the gretest mgnitude displcement during the time

More information

Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I

Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I Exm Stuy Guie Mth 2020 - Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

APPLICATION OF INTEGRALS

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

More information

Notes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams

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

More information

r 2 F ds W = r 1 qe ds = q

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

More information

The invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism.

The invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism. Instrutor: Longfei Li Mth 43 Leture Notes 16. Line Integrls The invention of line integrls is motivted by solving problems in fluid flow, fores, eletriity nd mgnetism. Line Integrls of Funtion We n integrte

More information

Binary Representation of Numbers Autar Kaw

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

More information

QUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution

QUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution QUADRATURE METHODS Kenneth L. Judd Hoover Institution July 19, 2011 1 Integrtion Most integrls cnnot be evluted nlyticlly Integrls frequently rise in economics Expected utility Discounted utility nd profits

More information

The Calculus of Variations: An Introduction. By Kolo Sunday Goshi

The Calculus of Variations: An Introduction. By Kolo Sunday Goshi The Clculus of Vritions: An Introduction By Kolo Sundy Goshi Some Greek Mythology Queen Dido of Tyre Fled Tyre fter the deth of her husbnd Arrived t wht is present dy Liby Irbs (King of Liby) offer Tell

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Surface Area and Volume

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

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

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

More information

6.5 - Areas of Surfaces of Revolution and the Theorems of Pappus

6.5 - Areas of Surfaces of Revolution and the Theorems of Pappus Lecture_06_05.n 1 6.5 - Ares of Surfces of Revolution n the Theorems of Pppus Introuction Suppose we rotte some curve out line to otin surfce, we cn use efinite integrl to clculte the re of the surfce.

More information

EQUATIONS OF LINES AND PLANES

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

More information

Two special Right-triangles 1. The

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

More information

Section 7-4 Translation of Axes

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

More information

Let us recall some facts you have learnt in previous grades under the topic Area.

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

More information

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS 4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

More information

SUBSTITUTION I.. f(ax + b)

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

More information

Ae2 Mathematics : Fourier Series

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

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

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.

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Jackson 2.23 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.23 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson.3 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: A hollow cube hs conducting wlls defined by six plnes x =, y =, z =, nd x =, y =, z =. The wlls z =

More information

Triple Integrals in Cylindrical or Spherical Coordinates

Triple Integrals in Cylindrical or Spherical Coordinates Triple Integrals in Clindrical or Spherical Coordinates. Find the volume of the solid ball 2 + 2 + 2. Solution. Let be the ball. We know b #a of the worksheet Triple Integrals that the volume of is given

More information

Chapter 6 Solving equations

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

More information

Square Roots Teacher Notes

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

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Brief review of prerequisites for ECON4140/4145

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.

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period: Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

More information

Applications of Integration Day 1

Applications of Integration Day 1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

More information

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc. Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of

More information

Theory of Forces. Forces and Motion

Theory of Forces. Forces and Motion his eek extbook -- Red Chpter 4, 5 Competent roblem Solver - Chpter 4 re-lb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics - Everything

More information

Pure C4. Revision Notes

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

More information

HCP crystal planes are described using the Miller-Bravais indices, (hkil) Draw the hexagonal crystal planes whose Miller-Bravais indices are:

HCP crystal planes are described using the Miller-Bravais indices, (hkil) Draw the hexagonal crystal planes whose Miller-Bravais indices are: .6 How re crystllogrphic plnes indicted in HCP unit cells? In HCP unit cells, crystllogrphic plnes re indicted using four indices which correspond to four xes: three bsl xes of the unit cell,,, nd, which

More information

1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s

1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s Crete ssignment, 99552, Homework 5, Sep 15 t 10:11 m 1 This print-out should he 30 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. The due time

More information

Exponential and Logarithmic Functions

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

More information