Volumes of Solids of Revolution. We revolve this curve about the x-axis and create a solid of revolution.
|
|
- Kellie O’Connor’
- 6 years ago
- Views:
Transcription
1 Volumes of Solids of Revolution Consider the function ( ) from a = to b = We revolve this curve about the x-axis and create a solid of revolution We want to find the volume of this solid. We begin by taking n cross-sectional slices of width.
2 Page The shape of one of these slices can be approximated by a disk. We will find the formula for the volume of one of these disks and then use Riemann sums and limits to produce an integral that will give us the exact volume of our solid of revolution. Laying a disk on its side, compute its volume as a cylinder. V = area of the base * height Note that for this disk, r is the distance between the x-axis and the curve. That is, for this disk, ( ) for some x in that slice. Likewise,. So the volume of this disk is ( ( )). Then when we approximate the volume of the solid of revolution using disks, the volume of the ith disk is ( ( )). The sum of the volumes of n disks is a Riemann Sum: ( ( )). Taking the limit as n goes to infinity, we get the integral ( ( )). Thus the integral for the volume of the solid of revolution generated by revolving the curve ( ) from x = a to x = b about the x-axis is ( ( )).
3 Page So, the volume of the solid of revolution that is generated by revolving the curve ( ) from a = to b = 9 about the x-axis is ( ). Ah, but what if we revolve ( ) about the y-axis? We will get to that on p. 5. But first, what if we revolve the region between two curves about the x-axis? For example, consider the region between ( ), y= and the y-axis When we revolve this about the x-axis, we will get something like a vase.
4 Page If we were to take cross-sectional slices, we would get something shaped like washers: disks with a smaller disk cut out of the center: The volume of a washer is the volume of the larger disk minus the volume of the inner disk that has been removed. Larger disk: V = area of base * height = where r =. Inner disk: V = area of base * height = where r =. So, the volume of this solid of revolution, using the method of washers is ( ) So, if ( ) ( ) on the interval [ ] then the volume of the solid of revolution generated by revolving the region enclosed by ( ) ( ) and is ( ( )) ( ( )) and, using properties of integrals, this is the same as (( ( )) ( ( )) )
5 Page 5 We now return to the question of what if we revolve ( ) about the y-axis: There are two possible ways to approach this. If ( ) is a one-to-one function on [a, b] and we can write x as a function of y, we can still use the method of disks, but our cross-sectional slicing is horizontal. For each slice, the approximating disk s radius is the distance from the y-axis to the curve: ( ) and the thickness of the disk is dy. The variable y is going from ( ) to ( ), so the volume is ( ( )) In general, using disks when revolving ( ) about the y-axis: ( ( )) where c and d are the lower and upper limits on y. In the problem above, revolving our function from a = to b = 9 about the y-axis, we would solve for x, giving us, then change our limits of integration from and b and and d: Lower limit: ( ) Upper limit: ( ). ( ). But if ( ) is not a one-to-one function on [a, b] or if we cannot write x as a function of y, or we are unable to evaluate the resulting integral, we cannot use the method of disks, so another method will have to be used.
6 Page 6 Using the solid from the previous example: This time, we take vertical cylindrical slices, sort of like taking the core out of a pineapple, except we take thin cylindrical slices We approximate these with perfectly cylindrical shells. When we examine one of these cylindrical shells, we see that we can cut it vertically and lay it flat as a rectangular slab. Volume of the cylindrical shell = volume of slab: The width of the rectangle is the circumference of the base of the cylinder:. Looking at the cylindrical slice while it is still inside the solid, we see that r = x. So, Next,., so. Finally,. So the volume of the cylindrical shell is ( ).
7 Page Take the Riemann sum of the volumes of n cylindrical shells: ( ) Then take the limit as n goes to infinity, and we have the integral ( ) = ( ) = So, using the method of cylindrical shells, ( )
Tuesday, September 29, Page 453. Problem 5
Tuesday, September 9, 15 Page 5 Problem 5 Problem. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by y = x, y = x 5 about the x-axis. Solution.
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationTechnique 1: Volumes by Slicing
Finding Volumes of Solids We have used integrals to find the areas of regions under curves; it may not seem obvious at first, but we can actually use similar methods to find volumes of certain types of
More informationLESSON 14: VOLUME OF SOLIDS OF REVOLUTION SEPTEMBER 27, 2017
LESSON 4: VOLUME OF SOLIDS OF REVOLUTION SEPTEMBER 27, 27 We continue to expand our understanding of solids of revolution. The key takeaway from today s lesson is that finding the volume of a solid of
More information8.1 Integral as Net Change
8.1 Integral as Net Change Key Concepts/Skill Objectives Chapter 8 Review Guide 1) Solve problems in which a rate is integrated to find the net change over time in a variety of Terms: applications Linear
More information5.5 Volumes: Tubes. The Tube Method. = (2π [radius]) (height) ( x k ) = (2πc k ) f (c k ) x k. 5.5 volumes: tubes 435
5.5 volumes: tubes 45 5.5 Volumes: Tubes In Section 5., we devised the disk method to find the volume swept out when a region is revolved about a line. To find the volume swept out when revolving a region
More informationVolume: The Disk Method. Using the integral to find volume.
Volume: The Disk Method Using the integral to find volume. If a region in a plane is revolved about a line, the resulting solid is a solid of revolution and the line is called the axis of revolution. y
More informationChapter 6 Some Applications of the Integral
Chapter 6 Some Applications of the Integral Section 6.1 More on Area a. Representative Rectangle b. Vertical Separation c. Example d. Integration with Respect to y e. Example Section 6.2 Volume by Parallel
More informationIntegration to Compute Volumes, Work. Goals: Volumes by Slicing Volumes by Cylindrical Shells Work
Week #8: Integration to Compute Volumes, Work Goals: Volumes by Slicing Volumes by Cylindrical Shells Work 1 Volumes by Slicing - 1 Volumes by Slicing In the integration problems considered in this section
More informationLecture 21. Section 6.1 More on Area Section 6.2 Volume by Parallel Cross Section. Jiwen He. Department of Mathematics, University of Houston
Section 6.1 Section 6.2 Lecture 21 Section 6.1 More on Area Section 6.2 Volume by Parallel Cross Section Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math1431
More informationf x and the x axis on an interval from x a and
Unit 6: Chapter 8 Areas and Volumes & Density Functions Part 1: Areas To find the area bounded by a function bwe use the integral: f d b a b 0 f d f d. a b a f and the ais on an interval from a and. This
More informationVolume of Solid of Known Cross-Sections
Volume of Solid of Known Cross-Sections Problem: To find the volume of a given solid S. What do we know about the solid? Suppose we are told what the cross-sections perpendicular to some axis are. Figure:
More informationChapter 7: Applications of Integration
Chapter 7: Applications of Integration Fall 214 Department of Mathematics Hong Kong Baptist University 1 / 21 7.1 Volumes by Slicing Solids of Revolution In this section, we show how volumes of certain
More informationReview: Exam Material to be covered: 6.1, 6.2, 6.3, 6.5 plus review of u, du substitution.
Review: Exam. Goals for this portion of the course: Be able to compute the area between curves, the volume of solids of revolution, and understand the mean value of a function. We had three basic volumes:
More informationMCB4UW Handout 7.6. Comparison of the Disk/Washer and Shell Methods. V f x g x. V f y g y
MCBUW Handout 7.6 Comparison of the Disk/Washer and Shell Methods Method Ais of Formula Notes aout the Revolution Representative Rectangle a Disk Method -ais V f d -ais a V g d Washer Method -ais a V f
More information8.2 APPLICATIONS TO GEOMETRY
8.2 APPLICATIONS TO GEOMETRY In Section 8.1, we calculated volumes using slicing and definite integrals. In this section, we use the same method to calculate the volumes of more complicated regions as
More informationHOMEWORK SOLUTIONS MATH 1910 Sections 6.1, 6.2, 6.3 Fall 2016
HOMEWORK SOLUTIONS MATH 191 Sections.1,.,. Fall 1 Problem.1.19 Find the area of the shaded region. SOLUTION. The equation of the line passing through ( π, is given by y 1() = π, and the equation of the
More informationd` = 1+( dy , which is part of the cone.
7.5 Surface area When we did areas, the basic slices were rectangles, with A = h x or h y. When we did volumes of revolution, the basic slices came from revolving rectangles around an axis. Depending on
More informationSolutions to Homework 1
Solutions to Homework 1 1. Let f(x) = x 2, a = 1, b = 2, and let x = a = 1, x 1 = 1.1, x 2 = 1.2, x 3 = 1.4, x 4 = b = 2. Let P = (x,..., x 4 ), so that P is a partition of the interval [1, 2]. List the
More informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
More informationSolid of Revolution. IB Mathematic HL. International Baccalaureate Program. October 31, 2016
1 Solid of Revolution IB Mathematic HL International Baccalaureate Program 2070 October 31, 2016 2 Introduction: Geometry is one of the areas of math that I feel passionate about, thereby I am inspired
More information14.1. Multiple Integration. Iterated Integrals and Area in the Plane. Iterated Integrals. Iterated Integrals. MAC2313 Calculus III - Chapter 14
14 Multiple Integration 14.1 Iterated Integrals and Area in the Plane Objectives Evaluate an iterated integral. Use an iterated integral to find the area of a plane region. Copyright Cengage Learning.
More informationVolumes of Solids of Revolution Lecture #6 a
Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply
More informationA = (cosh x sinh x) dx = (sinh x cosh x) = sinh1 cosh1 sinh 0 + cosh 0 =
Calculus 7 Review Consider the region between curves y= cosh, y= sinh, =, =.. Find the area of the region. e + e e e Solution. Recall that cosh = and sinh =, whence sinh cosh. Therefore the area is given
More informationMath 1b Midterm I Solutions Tuesday, March 14, 2006
Math b Midterm I Solutions Tuesday, March, 6 March 5, 6. (6 points) Which of the following gives the area bounded on the left by the y-axis, on the right by the curve y = 3 arcsin x and above by y = 3π/?
More informationUnit #13 - Integration to Find Areas and Volumes, Volumes of Revolution
Unit #1 - Integration to Find Areas and Volumes, Volumes of Revolution Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Areas In Questions #1-8, find the area of one strip
More informationArnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment VMultIntegrals1Double due 04/03/2008 at 02:00am EST.
WeBWorK assignment VMultIntegralsouble due 04/03/2008 at 02:00am ST.. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8.pg Consider the solid that lies above the square = [0,2] [0,2] and below the
More informationChapter 6: Applications of Integration
Chapter 6: Applications of Integration Section 6.3 Volumes by Cylindrical Shells Sec. 6.3: Volumes: Cylindrical Shell Method Cylindrical Shell Method dv = 2πrh thickness V = න a b 2πrh thickness Thickness
More informationCalculus II. Philippe Rukimbira. Department of Mathematics Florida International University PR (FIU) MAC / 1
Calculus II Philippe Rukimbira Department of Mathematics Florida International University PR (FIU) MAC 2312 1 / 1 5.4. Sigma notation; The definition of area as limit Assignment: page 350, #11-15, 27,
More informationMath 75B Practice Midterm III Solutions Chapter 6 (Stewart) Multiple Choice. Circle the letter of the best answer.
Math 75B Practice Midterm III Solutions Chapter 6 Stewart) English system formulas: Metric system formulas: ft. = in. F = m a 58 ft. = mi. g = 9.8 m/s 6 oz. = lb. cm = m Weight of water: ω = 6.5 lb./ft.
More informationPhysics 9 WS E3 (rev. 1.0) Page 1
Physics 9 WS E3 (rev. 1.0) Page 1 E-3. Gauss s Law Questions for discussion 1. Consider a pair of point charges ±Q, fixed in place near one another as shown. a) On the diagram above, sketch the field created
More informationHAND IN PART. Prof. Girardi Math 142 Spring Exam 3 PIN:
HAND IN PART Prof. Girardi Math 142 Spring 2014 04.17.2014 Exam 3 MARK BOX problem points possible your score 0A 9 0B 8 0C 10 0D 12 NAME: PIN: solution key Total for 0 39 Total for 1 10 61 % 100 INSTRUCTIONS
More informationand y c from x 0 to x 1
Math 44 Activity 9 (Due by end of class August 6). Find the value of c, c, that minimizes the volume of the solid generated by revolving the region between the graphs of y 4 and y c from to about the line
More informationExamples from Section 6.3: Volume by Cylindrical Shells Page 1
Examples from Section 6.3: Volume by Cylindrical Shells Page 1 Questions Example Find the volume of the region which is created when we rotate the region below y = x x 3 and above < x < about the y-axis.
More informationExam 3. MA 114 Exam 3 Fall Multiple Choice Questions. 1. Find the average value of the function f (x) = 2 sin x sin 2x on 0 x π. C. 0 D. 4 E.
Exam 3 Multiple Choice Questions 1. Find the average value of the function f (x) = sin x sin x on x π. A. π 5 π C. E. 5. Find the volume of the solid S whose base is the disk bounded by the circle x +
More informationPROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH SECTION 6.5
1 MATH 16A LECTURE. DECEMBER 9, 2008. PROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. I HOPE YOU ALL WILL MISS IT AS MUCH AS I DO. SO WHAT I WAS PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH
More informationProblem Solving 3: Calculating the Electric Field of Highly Symmetric Distributions of Charge Using Gauss s Law
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Problem Solving 3: Calculating the Electric Field of Highly Symmetric Distributions of Charge Using Gauss s Law REFERENCE: Section 4.2, 8.02
More information8.4 Density and 8.5 Work Group Work Target Practice
8.4 Density and 8.5 Work Group Work Target Practice 1. The density of oil in a circular oil slick on the surface of the ocean at a distance meters from the center of the slick is given by δ(r) = 5 1+r
More informationMATH 101 COURSE SYLLABUS
TOPICS OF THE COURSE 2. LIMITS AND RATE OF CHANGE : (8 Hours) Introduction to Limits, Definition of Limit, Techniques for Finding Limits, Limits Involving Infinity, Continuous functions. 3. THE DERIVATIVE
More informationMath 122 Fall Handout 15: Review Problems for the Cumulative Final Exam
Math 122 Fall 2008 Handout 15: Review Problems for the Cumulative Final Exam The topics that will be covered on Final Exam are as follows. Integration formulas. U-substitution. Integration by parts. Integration
More informationAP Physics C. Gauss s Law. Free Response Problems
AP Physics Gauss s Law Free Response Problems 1. A flat sheet of glass of area 0.4 m 2 is placed in a uniform electric field E = 500 N/. The normal line to the sheet makes an angle θ = 60 ẘith the electric
More informationHarbor Creek School District
Unit 1 Days 1-9 Evaluate one-sided two-sided limits, given the graph of a function. Limits, Evaluate limits using tables calculators. Continuity Evaluate limits using direct substitution. Differentiability
More information5) Two large metal plates are held a distance h apart, one at a potential zero, the other
Promlems 1) Find charge distribution on a grounded conducting sphere with radious R centered at the origin due to a charge q at a position (r,θ,φ) outside of the sphere. Plot the charge distribution as
More informationMATH 230 CALCULUS II OVERVIEW
MATH 230 CALCULUS II OVERVIEW This overview is designed to give you a brief look into some of the major topics covered in Calculus II. This short introduction is just a glimpse, and by no means the whole
More informationMidterm Exam #1. (y 2, y) (y + 2, y) (1, 1)
Math 5B Integral Calculus March 7, 7 Midterm Eam # Name: Answer Key David Arnold Instructions. points) This eam is open notes, open book. This includes any supplementary tets or online documents. You are
More informationAnswer Key. ( 1) n (2x+3) n. n n=1. (2x+3) n. = lim. < 1 or 2x+3 < 4. ( 1) ( 1) 2n n
Math Midterm Eam #3 December, 3 Answer Key. [5 Points] Find the Interval and Radius of Convergence for the following power series. Analyze carefully and with full justification. Use Ratio Test. L lim a
More informationAnswers for Ch. 6 Review: Applications of the Integral
Answers for Ch. 6 Review: Applications of the Integral. The formula for the average value of a function, which you must have stored in your magical mathematical brain, is b b a f d. a d / / 8 6 6 ( 8 )
More informationSample Final Questions: Solutions Math 21B, Winter y ( y 1)(1 + y)) = A y + B
Sample Final Questions: Solutions Math 2B, Winter 23. Evaluate the following integrals: tan a) y y dy; b) x dx; c) 3 x 2 + x dx. a) We use partial fractions: y y 3 = y y ) + y)) = A y + B y + C y +. Putting
More informationMath 76 Practice Problems for Midterm II Solutions
Math 76 Practice Problems for Midterm II Solutions 6.4-8. DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You may expect to
More informationt 2 + 2t dt = (t + 1) dt + 1 = arctan t x + 6 x(x 3)(x + 2) = A x +
MATH 06 0 Practice Exam #. (0 points) Evaluate the following integrals: (a) (0 points). t +t+7 This is an irreducible quadratic; its denominator can thus be rephrased via completion of the square as a
More informationApplications of integrals
ApplicationsofIntegrals.nb Applications of integrals There are many applications of definite integrals and we cannot include all of them in a document intended to be a review. However there are some very
More informationDouble Integrals. P. Sam Johnson. February 4, P. Sam Johnson (NIT Karnataka) (NIT Karnataka) Double Integrals February 4, / 57
Double Integrals P. Sam Johnson February 4, 2018 P. Sam Johnson (NIT Karnataka) (NIT Karnataka) Double Integrals February 4, 2018 1 / 57 Overview We defined the definite integral of a continuous function
More informationHOMEWORK SOLUTIONS MATH 1910 Sections 6.4, 6.5, 7.1 Fall 2016
HOMEWORK SOLUTIONS MATH 9 Sections 6.4, 6.5, 7. Fall 6 Problem 6.4. Sketch the region enclosed by x = 4 y +, x = 4y, and y =. Use the Shell Method to calculate the volume of rotation about the x-axis SOLUTION.
More informationChapter 22 Gauss s Law. Copyright 2009 Pearson Education, Inc.
Chapter 22 Gauss s Law 22-1 Electric Flux Electric flux: Electric flux through an area is proportional to the total number of field lines crossing the area. 22-1 Electric Flux Example 22-1: Electric flux.
More informationHUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK
HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT A P C a l c u l u s ( B C ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS Limits and Continuity Derivatives
More informationVector Calculus. Dr. D. Sukumar. February 1, 2016
Vector Calculus Dr. D. Sukumar February 1, 2016 Green s Theorem Tangent form or Ciculation-Curl form c Mdx + Ndy = R ( N x M ) da y Green s Theorem Tangent form or Ciculation-Curl form Stoke s Theorem
More informationMath 140 Final Sample A Solutions. Tyrone Crisp
Math 4 Final Sample A Solutions Tyrone Crisp (B) Direct substitution gives, so the limit is infinite. When is close to, but greater than,, the numerator is negative while the denominator is positive. So
More informationPre-Algebra Homework 10 Geometry: Solutions
Pre-Algebra Homework 10 Geometry: Solutions Use π = 3 for your calculations 1. I am building a ramp. The length of the ramp along the ground is 12 meters and it s 5 meters high and 5 meters wide. I want
More informationThe base of each cylinder is called a cross-section.
6. Volume y Slicing Gol: To find the volume of olid uing econd emeter clculu Volume y Cro-Section Volume y Dik Volume y Wher Volume y Slicing Volume y Shell 6. Volume y Slicing 6. Volume y Slicing Gol:
More informationNOT FOR SALE 6 APPLICATIONS OF INTEGRATION. 6.1 Areas Between Curves. Cengage Learning. All Rights Reserved. ( ) = 4 ln 1 = 45.
6 APPLICATIONS OF INTEGRATION 6. Areas Between Curves... 4. 8 ( ) 8 8 4 4 ln ( ln 8) 4 ln 45 4 ln 8 ( ) ( ) ( ) + + + + + ( ) ( 4) ( +6) + ( 8 + 7) 9 5. ( ) + ( +) ( + ) + 4 6. ( sin ) +cos 8 8 + 7. The
More informationRectangular box of sizes (dimensions) w,l,h wlh Right cylinder of radius r and height h r 2 h
Volumes: Slicing Method, Method of Disks and Washers -.,.. Volumes of Some Regular Solids: Solid Volume Rectangular bo of sizes (dimensions) w,l,h wlh Right clinder of radius r and height h r h Right cone
More information1993 AP Calculus AB: Section I
99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among
More informationMITOCW ocw-18_02-f07-lec25_220k
MITOCW ocw-18_02-f07-lec25_220k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More information(a) Use washer cross sections: a washer has
Section 8 V / (sec tan ) / / / / [ tan sec tan ] / / (sec sec tan + tan ) + tan sec / + A( ) s w (sec tan ), and / V (sec tan ), which b / same method as in part (a) equals A cross section has width w
More informationMTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation
MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation Lecture No 23 to 45 Complete and Important Question and answer 1. What is the difference between
More informationTest 7 wersja angielska
Test 7 wersja angielska 7.1A One revolution is the same as: A) 1 rad B) 57 rad C) π/2 rad D) π rad E) 2π rad 7.2A. If a wheel turns with constant angular speed then: A) each point on its rim moves with
More informationChapter 22 Gauss s Law. Copyright 2009 Pearson Education, Inc.
Chapter 22 Gauss s Law Electric Flux Gauss s Law Units of Chapter 22 Applications of Gauss s Law Experimental Basis of Gauss s and Coulomb s Laws 22-1 Electric Flux Electric flux: Electric flux through
More informationAP Calculus AB FIVES Sheet F = Fun at Home, I = Intellectual, V = Variety, E = Endeavors, S = Specimens
AP Calculus AB F heet F = Fun at Home, = ntellectual, = ariety, = ndeavors, = pecimens #62 Monday 11/26 F Worksheet 43 Print Worksheet 44 1) Find c for the Mean alue Theorem for 2) Find c for Rolle s Theorem
More informationChapter 23: Gauss Law. PHY2049: Chapter 23 1
Chapter 23: Gauss Law PHY2049: Chapter 23 1 Two Equivalent Laws for Electricity Coulomb s Law equivalent Gauss Law Derivation given in Sec. 23-5 (Read!) Not derived in this book (Requires vector calculus)
More informationApplied Calculus I. Lecture 36
Applied Calculus I Lecture 36 Computing the volume Consider a continuous function over an interval [a, b]. y a b x Computing the volume Consider a continuous function over an interval [a, b]. y y a b x
More informationAP Calculus BC Syllabus Course Overview
AP Calculus BC Syllabus Course Overview Textbook Anton, Bivens, and Davis. Calculus: Early Transcendentals, Combined version with Wiley PLUS. 9 th edition. Hoboken, NJ: John Wiley & Sons, Inc. 2009. Course
More informationRight Circular Cylinders A right circular cylinder is like a right prism except that its bases are congruent circles instead of congruent polygons.
Volume-Lateral Area-Total Area page #10 Right Circular Cylinders A right circular cylinder is like a right prism except that its bases are congruent circles instead of congruent polygons. base height base
More informationMath 132 Information for Test 2
Math 13 Information for Test Test will cover material from Sections 5.6, 5.7, 5.8, 6.1, 6., 6.3, 7.1, 7., and 7.3. The use of graphing calculators will not be allowed on the test. Some practice questions
More informationfree space (vacuum) permittivity [ F/m]
Electrostatic Fields Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charge distributions. The mathematical definition of the electrostatic field is derived
More informationLecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay. Poisson s and Laplace s Equations
Poisson s and Laplace s Equations Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We will spend some time in looking at the mathematical foundations of electrostatics.
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.13 INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) by A.J.Hobson 13.13.1 Introduction 13.13. The second moment of a volume of revolution about the y-axis 13.13.3
More informationAP Calculus. (Mr. Surowski) Homework from Chapter 7 (and some of Chapter 8)
AP Calculus (Mr. Surowski) Homework from Chapter 7 (and some of Chapter 8) Lesson 30 Integral as accumulation (7.):, 3, 5, 8 0, 7, 20 22, 25 (to do quadratic regression on our (TI-84 calculators, refer
More informationConcentric Circles Puzzle
In the image above, the inner circle has a circumference of 10 and the distance between the inner and outer circles is 3. If the circumference of the inner circle is increased to 11, and the distance between
More informationUpon completion of this course, the student should be able to satisfy the following objectives.
Homework: Chapter 6: o 6.1. #1, 2, 5, 9, 11, 17, 19, 23, 27, 41. o 6.2: 1, 5, 9, 11, 15, 17, 49. o 6.3: 1, 5, 9, 15, 17, 21, 23. o 6.4: 1, 3, 7, 9. o 6.5: 5, 9, 13, 17. Chapter 7: o 7.2: 1, 5, 15, 17,
More informationVarberg 8e-9e-ET Version Table of Contents Comparisons
Varberg 8e-9e-ET Version Table of Contents Comparisons 8th Edition 9th Edition Early Transcendentals 9 Ch Sec Title Ch Sec Title Ch Sec Title 1 PRELIMINARIES 0 PRELIMINARIES 0 PRELIMINARIES 1.1 The Real
More informationINGENIERÍA EN NANOTECNOLOGÍA
ETAPA DISCIPLINARIA TAREAS 385 TEORÍA ELECTROMAGNÉTICA Prof. E. Efren García G. Ensenada, B.C. México 206 Tarea. Two uniform line charges of ρ l = 4 nc/m each are parallel to the z axis at x = 0, y = ±4
More informationMATH 162. Midterm Exam 1 - Solutions February 22, 2007
MATH 62 Midterm Exam - Solutions February 22, 27. (8 points) Evaluate the following integrals: (a) x sin(x 4 + 7) dx Solution: Let u = x 4 + 7, then du = 4x dx and x sin(x 4 + 7) dx = 4 sin(u) du = 4 [
More informationWeek #15 - Word Problems & Differential Equations Section 8.2
Week #1 - Word Problems & Differential Equations Section 8. From Calculus, Single Variable by Huges-Hallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission
More information5 Integrals reviewed Basic facts U-substitution... 5
Contents 5 Integrals reviewed 5. Basic facts............................... 5.5 U-substitution............................. 5 6 Integral Applications 0 6. Area between two curves.......................
More information(Refer Slide Time: 2:08 min)
Applied Mechanics Prof. R. K. Mittal Department of Applied Mechanics Indian Institute of Technology, Delhi Lecture No. 11 Properties of Surfaces (Contd.) Today we will take up lecture eleven which is a
More informationChapter 6 Notes, Stewart 8e
Contents 6. Area between curves........................................ 6.. Area between the curve and the -ais.......................... 6.. Overview of Area of a Region Between Two Curves...................
More informationLearning Objectives for Math 166
Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the
More informationHUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK
HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT A P C a l c u l u s ( A B ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS Limits and Continuity Derivatives
More informationThe GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
More information5 Integrals reviewed Basic facts U-substitution... 4
Contents 5 Integrals reviewed 5. Basic facts............................... 5.5 U-substitution............................. 4 6 Integral Applications 0 6. Area between two curves.......................
More informationProof of the Equivalent Area of a Circle and a Right Triangle with Leg Lengths of the Radius and Circumference
Proof of the Equivalent Area of a ircle and a Right Triangle with Leg Lengths of the Radius and ircumference Brennan ain July 22, 2018 Abstract In this paper I seek to prove Archimedes Theorem that a circle
More informationENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-01
ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-01 3. Multiple Integration This chapter provides only a very brief introduction to the major topic of multiple integration. Uses of multiple
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 8. Single Variable Calculus Fall 6 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 3 8. Fall 6 Lecture 3:
More informationFor the intersections: cos x = 0 or sin x = 1 2
Chapter 6 Set-up examples The purpose of this document is to demonstrate the work that will be required if you are asked to set-up integrals on an exam and/or quiz.. Areas () Set up, do not evaluate, any
More informationENGI Parametric Vector Functions Page 5-01
ENGI 3425 5. Parametric Vector Functions Page 5-01 5. Parametric Vector Functions Contents: 5.1 Arc Length (Cartesian parametric and plane polar) 5.2 Surfaces of Revolution 5.3 Area under a Parametric
More informationTopic Outline AP CALCULUS AB:
Topic Outline AP CALCULUS AB: Unit 1: Basic tools and introduction to the derivative A. Limits and properties of limits Importance and novelty of limits Traditional definitions of the limit Graphical and
More informationDepartment of Mathematical 1 Limits. 1.1 Basic Factoring Example. x 1 x 2 1. lim
Contents 1 Limits 2 1.1 Basic Factoring Example...................................... 2 1.2 One-Sided Limit........................................... 3 1.3 Squeeze Theorem..........................................
More informationSingle Variable Calculus, Early Transcendentals
Single Variable Calculus, Early Transcendentals 978-1-63545-100-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax
More information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More informationPHY752, Fall 2016, Assigned Problems
PHY752, Fall 26, Assigned Problems For clarification or to point out a typo (or worse! please send email to curtright@miami.edu [] Find the URL for the course webpage and email it to curtright@miami.edu
More information