Integrals in cylindrical, spherical coordinates (Sect. 15.7)
|
|
- Marsha Bryan
- 7 years ago
- Views:
Transcription
1 Integrals in clindrical, spherical coordinates (Sect. 15.7) Integration in clindrical coordinates. eview: Polar coordinates in a plane. Clindrical coordinates in space. Triple integral in clindrical coordinates. Net class: Integration in spherical coordinates. eview: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates. eview: Polar coordinates in plane Definition The polar coordinates of a point P is the ordered pair (r, θ) defined b the picture. r P = ( r, ) = (,) Theorem (Cartesian-polar transformations) The Cartesian coordinates of a point P = (r, θ) are given b = r cos(θ), = r sin(θ). The polar coordinates of a point P = (, ) in the first and fourth quadrant are r = ( ) +, θ = arctan.
2 ecall: Polar coordinates in a plane Eample Epress in polar coordinates the integral I = Solution: ecall: = r cos(θ) and = r sin(θ). More often than not helps to sketch the integration region. The outer integration limit: [, ]. d d. = Then, for ever [, ] the coordinate satisfies [, ]. = The upper limit for is the curve =. Now is simple to describe this domain in polar coordinates: The line = is θ = π/4; the line = is θ 1 = π/. ecall: Polar coordinates in a plane Eample Epress in polar coordinates the integral I = d d. Solution: ecall: = r cos(θ), = r sin(θ), θ = π/4, θ 1 = π/. The lower integration limit in r is r =. = = r sin ( ) The upper integration limit is =, that is, = = r sin(θ). Hence r = / sin(θ) We conclude: d d = π/ / sin(θ) π/4 r cos(θ)(r dr) dθ.
3 Integrals in clindrical, spherical coordinates (Sect. 15.7) Integration in clindrical coordinates. eview: Polar coordinates in a plane. Clindrical coordinates in space. Triple integral in clindrical coordinates. Clindrical coordinates in space Definition The clindrical coordinates of a point P 3 is the ordered triple (r, θ, ) defined b the picture. emark: Clindrical coordinates are just polar coordinates on the plane = together with the vertical coordinate. Theorem (Cartesian-clindrical transformations) The Cartesian coordinates of a point P = (r, θ, ) in the first quadrant are given b = r cos(θ), = r sin(θ), and =. The clindrical coordinates of a point P = (,, ) in the first quadrant are given b r = +, θ = arctan(/), and =. r P
4 Clindrical coordinates in space Eample Use clindrical coordinates to describe the region = {(,, ) : + ( 1) 1, + }. Solution: We first sketch the region. The base of the region is at =, given b the disk + ( 1) 1. The top of the region is the paraboloid = +. In clindrical coordinates: = + = r, and = ( 1) = r r sin(θ) r sin(θ) Hence: = {(r, θ, ) : θ [, π], r [, sin(θ)], [, r ]}. Integrals in clindrical, spherical coordinates (Sect. 15.7) Integration in clindrical coordinates. eview: Polar coordinates in a plane. Clindrical coordinates in space. Triple integral in clindrical coordinates.
5 Triple integrals using clindrical coordinates Theorem If the function f : 3 is continuous, then the triple integral of function f in the region can be epressed in clindrical coordinates as follows, f dv = f (r, θ, ) r dr dθ d. emark: Clindrical coordinates are useful when the integration region is described in a simple wa using clindrical coordinates. Notice the etra factor r on the right-hand side. Triple integrals using clindrical coordinates Eample Find the volume of a clinder of radius and height h. Solution: = {(r, θ, ) : θ [, π], r [, ], [, h]}. V = π h d (r dr) dθ, h V = h π V = h π r dr dθ, dθ, We conclude: V = π h. V = h π,
6 Triple integrals using clindrical coordinates Eample Find the volume of a cone of base radius and height h. { Solution: = θ [, π], r [, ], [, h ]} r + h. h We conclude: V = 1 3 π h. V = π h(1 r/) V = h V = h π π d (r dr) dθ, ( 1 r ) r dr dθ, (r r ( V = h 3 ) π 3 ) dr dθ, dθ = πh 1 6. Triple integrals using clindrical coordinates Eample Sketch the region with volume V = π/ 9 r rd dr dθ. Solution: The integration region is = {(r, θ, ) : θ [, π/], r [, ], [, 9 r ]}. We upper boundar is a sphere, = 9 r + + = 3. The upper limit for r is r =, so 3 = 9 r 5 = 9 =
7 Triple integrals using clindrical coordinates Eample Change the integration order and compute the integral I = π 3 /3 r 3 dr d dθ. Solution: First, sketch the integration region. Start from the outer limits to the inner limits. r = /3 3 I = π 1 3 V = π V = π 3r 1 ( 3 3r 1 d r 3 dr dθ ) r 3 dr 3(r 3 r 4 ) dr 1 ( r 4 V = 6π 4 r 5 ) 1 5. So, I = 6π 1 3π, that is, I = 1. Triple integrals using clindrical coordinates Eample Find the centroid vector r =,, of the region in space = {(,, ) : +, + 4}. Solution: = + The smmetr of the region implies = and =. (We verif this result later on.) We onl need to compute. Since = 1 dv, we start V computing the total volume V. 4 We use clindrical coordinates. V = π 4 r d rdr dθ = π ( 4 )rdr = π r (4r r 3 )dr.
8 Triple integrals using clindrical coordinates Eample Find the centroid vector r =,, of the region in space = {(,, ) : +, + 4}. Solution: V = π [ ( r (4r r 3 ) dr = π 4 ) ( r 4 4 Hence V = π(8 4), so V = 8π. Then, is given b, = 1 8π = 1 8 π 4 = 1 8 r d rdr dθ = π 8π )]. ( 4 ) rdr; r (16r r 5 ) dr = 1 [ ( r 16 ) ( r ( ) = = 8 3. )] ; Triple integrals using clindrical coordinates Eample Find the centroid vector r =,, of the region in space = {(,, ) : +, + 4}. Solution: We obtained = 8 3. It is simple to see that = and =. For eample, π 4 = 1 [ ] r cos(θ) d rdr dθ 8π r = 1 [ π ][ 4 cos(θ)dθ d r dr 8π r ]. But π cos(θ)dθ = sin(π) sin() =, so =. A similar calculation shows =. Hence r =,, 8/3.
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 informationDouble 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 informationMULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then
MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationArea and Arc Length in Polar Coordinates
Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More informationAB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss
AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =
More informationM PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM
68 Theor Supplement Section M M POOF OF THE DIEGENE THEOEM ND STOKES THEOEM In this section we give proofs of the Divergence Theorem Stokes Theorem using the definitions in artesian coordinates. Proof
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationv 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)
0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3-space. This time the outcome will be a vector in 3-space. Definition
More informationComplex Numbers. w = f(z) z. Examples
omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
More informationAffine Transformations
A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates
More informationSection 9.1 Vectors in Two Dimensions
Section 9.1 Vectors in Two Dimensions Geometric Description of Vectors A vector in the plane is a line segment with an assigned direction. We sketch a vector as shown in the first Figure below with an
More informationFundamental Theorems of Vector Calculus
Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use
More informationLecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...
CONTENTS i Contents Lecture Introduction. Rectangular Coordinate Sstems..................... Vectors.................................. 3 Lecture Length, Dot Product, Cross Product 5. Length...................................
More informationReview B: Coordinate Systems
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B-2 B.1.1 Infinitesimal Line Element... B-4 B.1.2 Infinitesimal Area Element...
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationSolutions - Homework sections 17.7-17.9
olutions - Homework sections 7.7-7.9 7.7 6. valuate xy d, where is the triangle with vertices (,, ), (,, ), and (,, ). The three points - and therefore the triangle between them - are on the plane x +
More informationSolutions for Math 311 Assignment #1
Solutions for Math 311 Assignment #1 (1) Show that (a) Re(iz) Im(z); (b) Im(iz) Re(z). Proof. Let z x + yi with x Re(z) and y Im(z). Then Re(iz) Re( y + xi) y Im(z) and Im(iz) Im( y + xi) x Re(z). () Verify
More informationTHE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second
More informationVector Calculus Solutions to Sample Final Examination #1
Vector alculus s to Sample Final Examination #1 1. Let f(x, y) e xy sin(x + y). (a) In what direction, starting at (,π/), is f changing the fastest? (b) In what directions starting at (,π/) is f changing
More informationex) What is the component form of the vector shown in the picture above?
Vectors A ector is a directed line segment, which has both a magnitude (length) and direction. A ector can be created using any two points in the plane, the direction of the ector is usually denoted by
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationEE2 Mathematics : Vector Calculus
EE2 Mathematics : Vector alculus J. D. Gibbon Professor J. D Gibbon, Dept of Mathematics) j.d.gibbon@ic.ac.uk http://www2.imperial.ac.uk/ jdg These notes are not identical word-for-word with m lectures
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationReview A: Vector Analysis
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A-0 A.1 Vectors A-2 A.1.1 Introduction A-2 A.1.2 Properties of a Vector A-2 A.1.3 Application of Vectors
More informationMath 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationReview Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
More informationStokes' Theorem Examples
Stokes' Theorem Eamples Stokes' Theorem relates surface integrals and line integrals. STOKES' TEOREM Let F be a vector field. Let be an oriented surface, and let be the boundar curve of, oriented using
More informationChapter 6 Circular Motion
Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations
More informationMagnetic Dipoles. Magnetic Field of Current Loop. B r. PHY2061 Enriched Physics 2 Lecture Notes
Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to
More information10 Polar Coordinates, Parametric Equations
Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates
More informationGradient, Divergence and Curl in Curvilinear Coordinates
Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent
More informationChapter 9 Circular Motion Dynamics
Chapter 9 Circular Motion Dynamics 9. Introduction Newton s Second Law and Circular Motion... 9. Universal Law of Gravitation and the Circular Orbit of the Moon... 9.. Universal Law of Gravitation... 3
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationMath 2443, Section 16.3
Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationSECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11.
SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationIntroduction to polarization of light
Chapter 2 Introduction to polarization of light This Chapter treats the polarization of electromagnetic waves. In Section 2.1 the concept of light polarization is discussed and its Jones formalism is presented.
More informationAddition and Subtraction of Vectors
ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b
More informationTeacher Page. 1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image.
Teacher Page Geometr / Da # 10 oordinate Geometr (5 min.) 9-.G.3.1 9-.G.3.2 9-.G.3.3 9-.G.3. Use rigid motions (compositions of reflections, translations and rotations) to determine whether two geometric
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationSection 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationFifty Famous Curves, Lots of Calculus Questions, And a Few Answers
Department of Mathematics, Computer Science, and Statistics Bloomsburg Universit Bloomsburg, Pennslvania 785 Fift Famous Curves, Lots of Calculus Questions, And a Few Answers Summar Sophisticated calculators
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationPlane Stress Transformations
6 Plane Stress Transformations 6 1 Lecture 6: PLANE STRESS TRANSFORMATIONS TABLE OF CONTENTS Page 6.1 Introduction..................... 6 3 6. Thin Plate in Plate Stress................ 6 3 6.3 D Stress
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationThe Two-Body Problem
The Two-Body Problem Abstract In my short essay on Kepler s laws of planetary motion and Newton s law of universal gravitation, the trajectory of one massive object near another was shown to be a conic
More informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationMAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:
MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex
More informationMATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 3 - NUMERICAL INTEGRATION METHODS
MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL - NUMERICAL INTEGRATION METHODS This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationVersion 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence
Version 005 Eam Review Practice Problems NOT FOR A GRADE aleander 5575 This print-out should have 47 questions Multiple-choice questions may continue on the net column or page find all choices before answering
More informationComplex Algebra. What is the identity, the number such that it times any number leaves that number alone?
Complex Algebra When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in some sense not real. Later, when probably one of the students of Pythagoras
More informationSupporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 2. Coordinate geometry
1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 1 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometr: Module Coordinate geometr Coordinate geometr A guide for teachers (Years
More informationEQUILIBRIUM STRESS SYSTEMS
EQUILIBRIUM STRESS SYSTEMS Definition of stress The general definition of stress is: Stress = Force Area where the area is the cross-sectional area on which the force is acting. Consider the rectangular
More informationCross Products and Moments of Force
4 Cross Products and Moments of Force Ref: Hibbeler 4.2-4.3, edford & Fowler: Statics 2.6, 4.3 In geometric terms, the cross product of two vectors, A and, produces a new vector, C, with a direction perpendicular
More informationFACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationVector surface area Differentials in an OCS
Calculus and Coordinate systems EE 311 - Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1
More informationUse Geometry Expressions to find equations of curves. Use Geometry Expressions to translate and dilate figures.
Learning Objectives Loci and Conics Lesson 2: The Circle Level: Precalculus Time required: 90 minutes Students are now acquainted with the idea of locus, and how Geometry Expressions can be used to explore
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More informationCHAPTER 29 VOLUMES AND SURFACE AREAS OF COMMON SOLIDS
CHAPTER 9 VOLUMES AND SURFACE AREAS OF COMMON EXERCISE 14 Page 9 SOLIDS 1. Change a volume of 1 00 000 cm to cubic metres. 1m = 10 cm or 1cm = 10 6m 6 Hence, 1 00 000 cm = 1 00 000 10 6m = 1. m. Change
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationThe Vector or Cross Product
The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero
More informationGiven three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);
1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the
More information