Volume: The Disk Method. Using the integral to find volume.
|
|
- Jeffery Hoover
- 6 years ago
- Views:
Transcription
1 Volume: The Disk Method Using the integral to find volume.
2 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 = x
3 Volume of a Solid of Revolution In order to find the volume of a solid of revolution, you must first look at the cross sections of the solid. y = x The Cross Section is a circle so: A = πr 2 What is the r? A x = π x 2
4 Volume of a Solid of Revolution Then after finding a formula for the area of the cross section, you will integrate this function to find the volume. y = x A x = π x 2 A x = πx b A(x) a dx 10 πx 0 dx
5 Simplifying the formula. Whenever we revolve the area around an axis of revolution, our cross sections are circles. Since our radius will be a function of x, we can rewrite R as R(x). V = a b V = π R x 2 dx a b A(x)dx b V = π R x 2 dx a A Region to revolve. B Y = R(x) Note that R(x) represents the radius of the revolved cross section.
6 Example 1: Revolve the area of y = 4 x 2 from x = 2 to x = 2 around the x axis. b V = π R x 2 dx a y = 4 x 2 R x = 4 x 2 V = π x 2 2 dx
7 Example 2: Find the volume formed by revolving the region f x = sin x about the x-axis from 0 to π.
8 The solid formed from f x = sin x
9 What is the volume of the solid formed by revolving about the y-axis from y = 0 to y = 4? Step One: determine the radius of a cross section. y = 16 x 2 y 2 = 16 x 2 x = 16 y 2 Step Two: Set up same formula and integrate with respect to y.
10 Homework Section 7.2 (1-4 and 7-10)
11
12
13
14
15
16 8.
17 9.
18
19 Day 2: Revolving about a line that is not the x or y axis. If you are revolving around a line that is parallel to the x axis, you will integrate with respect to x. If you are revolving around a line that is parallel to the y axis, you will integrate with respect to y.
20 What is your radius? When doing a volume problem, you need to ask What is the radius of the circle I am creating? When revolving about the Line y = 2.. F(x) R(1) = R x = y = 2 V = π F x 2 2 dx 3 3
21 Revolving about a line that is not the x-axis. Example: Find the volume of the solid formed by revolving the region bounded by f x = 2 x 2 and g x = 1 about the line y = 1. To determine the radius.. R x = f x g x = 2 x 2 1 = 1 x 2 If you let f x = g x, you can see that these two lines intersect at 1 and 1.
22
23 Revolving around Vertical line (not y-axis) What is the volume of the solid formed by revolving the lines x = 5 and x = y around the line x = 5? R y = y R y = y V = π 3 3 y dy
24 Homework Section 7.2 (6 (revolve around y=2), 11A, 11C, 12B, 12D, 16, 22,)
25 Revolve the region around Y = 2. 4 x2 4 = 2 x = ±2 2
26 V = π x dx V = π x dx V = π
27
28
29
30
31
32
33 Day 3: The Washer Method So far, all regions that we revolve have always been in contact with the access of revolution. What if we revolve a region that is not always in contact with the access of revolution?
34 The Washer Method The washer method is used to revolve a plane around an axis that it is not touching (either partly or entirely).
35 Imagine for a second that the rectangle touched the axis of revolution and you formed a solid cylinder as before. The volume would be π R 2 dx Now consider the area that is cut out from the cylinder. The radius of this empty cylinder is r. The volume of the smaller solid will be π r 2 dx If you subtract the smaller cylinder from the larger, you will have the volume. π R 2 dx π r 2 dx = π R 2 r 2 dx
36 V = π R x 2 r x 2 dx b a
37 Main Things to Consider in this type of Problem 1. IMAGINE YOU ARE REVOLVING A SOLID AREA THAT ALWAYS TOUCHES AXIS IN ORDER TO FIND R(X). 2. THINK ABOUT THE SMALLER AREA THAT IS CUT OUT IN ORDER TO FIND THE r(x). V = π R x 2 r x 2 dx b a
38 Set up this integral.. Revolve the area around the x-axis. V = π R x 2 r x 2 dx b a
39 Example 3 (Washer Method) Find the volume of the solid formed by revolving the region bounded by the graphs of y = x and y = x 2 about the x- axis.
40
41 y = x and x=4 revolved around x=7. x = y 2 R y = 7 y 2 6 r y = 7 4 = 3 2 π 7 y dy
42 y = 2x 2 and x = 2 and x axis around y = 9. R x = 9 r x = 9 2x 2 2 π x dx
43 Revolve the area enclosed by the above Graphs around the line y = 8. R x = 8 (x 2 + 1) r x = 8 ( x 2 + 2x + 5) 2 π 7 x x 2 2x dx
44 What if we tried to revolve around the line x = 8??? What would be the problem we would encounter?
45 Questions to ask yourself when solving. 1. Is the axis of revolution parallel to X axis dx or Y axis dy? 2. What is your solid/big radius? R X 3. What is the radius of the portion you will cut out? r(x)
46 Homework Section 7.2 (11d, 12c, 13a, 13b, 14a, 14b, 17)
47 11d.
48
49 13A. Revolve around X-axis
50
51 13b.
52
53 14A. Revolve around the X-Axis
54
55 14B.
56
57 17. y = 1 1+x, y = 0, x = 0, x = 3 Revolve around y = 4
58
59
60
61 After discussing these homework problems give students the following AP problems AB Free Response # AB Free Response #1 *Use Calculus in Motion to explain solutions.*
62
63
64 Summary Revolving around the x axis or y axis. Region always touches axis. Revolving around an axis parallel to x axis or y axis. Region always touches axis. Revolving around an axis that is not always in contact with region. Washer Method.
65 Area and Volume Review Set up the equations for revolving around the following lines? Y = 4 π 2 4 x 2 2 dx 2 X-Axis π x 2 2 dx 2 Y = 6 π 2 6 x dx 2
66 Area and Volume Review In words, tell me the difference between revolving around the following lines? A. Y = 4 B. X Axis C. Y = 6
67 Cross Sections and Volume In the following picture the axis of revolution is parallel to what? While the cross sections are perpendicular to what? If the cross sections are perpendicular to the x-axis... Volume = 0 1 A x dx **Note A(x) is a circle. Thus ** A x = π r x 2
68 In the following picture, The axis of revolution is parallel to what? The cross sections are perpendicular to what? If the cross sections are perpendicular to the y-axis Volume = 0 8 A y dy **Note A(y) is a circle. Thus ** A y = π r y 2
69 How would we find the volume of the following? Volume = 0 h A x dx Volume = h A y dy 0
70
71
72 Example: Find the volume of the solid whose base is bounded by the lines: f x = 1 x, g x = 1 + x and x = (The cross sections are perpendicular to the x-axis and form equilateral triangles). Area = base
73 The base will equal the difference in the Y values.
74 Homework *Write/Find better cross section homework for next year.* Section 7.2 (61, 62 A-C, odd) (Set up Equations ONLY!!) 2000 AP Test 1C 2003 AP (1) (Use Calculus in Motion to explain solutions)
75
76
77
78 After discussing 2000 AP Test 1C, 2003 AP (1). Do the following AP Problems AB 2007 (1) AB 2009 (4)
Technique 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 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 informationTuesday, 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 information6.1 Area Between Curves. Example 1: Calculate the area of the region between the parabola y = 1 x 2 and the line y = 1 x
AP Calculus 6.1 Area Between Curves Name: Goal: Calculate the Area between curves Keys to Success: Top Curve Bottom Curve (integrate w/respect to x or dx) Right Curve Left Curve (integrate w/respect to
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 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 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 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 informationAP Calculus AB. Review for Test: Applications of Integration
Name Review for Test: Applications of Integration AP Calculus AB Test Topics: Mean Value Theorem for Integrals (section 4.4) Average Value of a Function (manipulation of MVT for Integrals) (section 4.4)
More informationPrelim 1 Solutions V2 Math 1120
Feb., Prelim Solutions V Math Please show your reasoning and all your work. This is a 9 minute exam. Calculators are not needed or permitted. Good luck! Problem ) ( Points) Calculate the following: x a)
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 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 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 informationMATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS
MATH 181, FALL 17 - PROBLEM SET # 6 SOLUTIONS Part II (5 points) 1 (Thurs, Oct 6; Second Fundamental Theorem; + + + + + = 16 points) Let sinc(x) denote the sinc function { 1 if x =, sinc(x) = sin x if
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 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 informationFoundations of Calculus. November 18, 2014
Foundations of Calculus November 18, 2014 Contents 1 Conic Sections 3 11 A review of the coordinate system 3 12 Conic Sections 4 121 Circle 4 122 Parabola 5 123 Ellipse 5 124 Hyperbola 6 2 Review of Functions
More informationa) rectangles whose height is the right-hand endpoint b) rectangles whose height is the left-hand endpoint
CALCULUS OCTOBER 7 # Day Date Assignment Description 6 M / p. - #,, 8,, 9abc, abc. For problems 9 and, graph the function and approximate the integral using the left-hand endpoints, right-hand endpoints,
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 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 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 informationVolumes of Solids of Revolution. We revolve this curve about the x-axis and create a solid of revolution.
Volumes of Solids of Revolution Consider the function ( ) from a = to b = 9. 5 6 7 8 9 We revolve this curve about the x-axis and create a solid of revolution. - 5 6 7 8 9 - - - We want to find the volume
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing
More informationThe Definite Integral as an Accumulator
The Definite Integral as an Accumulator Louis A. Talman Department of Mathematical & Computer Sciences Metropolitan State College of Denver May, 5 In the last few years, the committee that writes the AP
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 informationGreen s Theorem. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Green s Theorem
Green s Theorem MATH 311, alculus III J. obert Buchanan Department of Mathematics Fall 2011 Main Idea Main idea: the line integral around a positively oriented, simple closed curve is related to a double
More informationMAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS
MAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT100 is a fast-paced and thorough tour of precalculus mathematics, where the choice of topics is primarily motivated by the conceptual and technical knowledge
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 informationQuiz 6 Practice Problems
Quiz 6 Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not
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 informationAP CALCULUS AB 2003 SCORING GUIDELINES
CORING GUIDELINE Question Let R be the shaded region bounded by the graphs of y the vertical line =, as shown in the figure above. = and y = e and (b) Find the volume of the solid generated when R is revolved
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 informationProblem Solving 1: The Mathematics of 8.02 Part I. Coordinate Systems
Problem Solving 1: The Mathematics of 8.02 Part I. Coordinate Systems In 8.02 we regularly use three different coordinate systems: rectangular (Cartesian), cylindrical and spherical. In order to become
More informationa) rectangles whose height is the right-hand endpoint b) rectangles whose height is the left-hand endpoint
# Day Date Assignment Description 36 M /3 p. 32-32 #, 2, 8, 2, 29abc, 3abc. For problems 29 and 3, graph the function and approximate the integral using the left-hand endpoints, right-hand endpoints, and
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing
More informationMAC 2311 Calculus I Spring 2004
MAC 2 Calculus I Spring 2004 Homework # Some Solutions.#. Since f (x) = d dx (ln x) =, the linearization at a = is x L(x) = f() + f ()(x ) = ln + (x ) = x. The answer is L(x) = x..#4. Since e 0 =, and
More informationCoffee Hour Problems and Solutions
Coffee Hour Problems and Solutions Edited by Matthew McMullen Fall 01 Week 1. Proposed by Matthew McMullen. Find all pairs of positive integers x and y such that 0x + 17y = 017. Solution. By inspection,
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 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 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 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 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 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 informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationLecture 10 - Moment of Inertia
Lecture 10 - oment of Inertia A Puzzle... Question For any object, there are typically many ways to calculate the moment of inertia I = r 2 dm, usually by doing the integration by considering different
More informationAP Calculus AB Mrs. Mills Carlmont High School
AP Calculus AB 015-016 Mrs. Mills Carlmont High School AP CALCULUS AB SUMMER ASSIGNMENT NAME: READ THE FOLLOWING DIRECTIONS CAREFULLY! Read through the notes & eamples for each page and then solve all
More informationQuestion. [The volume of a cone of radius r and height h is 1 3 πr2 h and the curved surface area is πrl where l is the slant height of the cone.
Q1 An experiment is conducted using the conical filter which is held with its axis vertical as shown. The filter has a radius of 10cm and semi-vertical angle 30. Chemical solution flows from the filter
More informationMath 113 (Calculus II) Final Exam KEY
Math (Calculus II) Final Exam KEY Short Answer. Fill in the blank with the appropriate answer.. (0 points) a. Let y = f (x) for x [a, b]. Give the formula for the length of the curve formed by the b graph
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 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 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 informationWebAssign Lesson 2-2 Volumes (Homework)
WebAssign Lesson 2-2 Volumes (Homework) Current Score : / 38 Due : Thursday, July 3 2014 11:00 AM MDT Jaimos Skriletz Math 175, section 31, Summer 2 2014 Instructor: Jaimos Skriletz 1. /2 points Find the
More informationPossible C4 questions from past papers P1 P3
Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
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 informationMathematics 1 Lecture Notes Chapter 1 Algebra Review
Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to
More informationCalculus II (Fall 2015) Practice Problems for Exam 1
Calculus II (Fall 15) Practice Problems for Exam 1 Note: Section divisions and instructions below are the same as they will be on the exam, so you will have a better idea of what to expect, though I will
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 informationMath 113 Winter 2005 Key
Name Student Number Section Number Instructor Math Winter 005 Key Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems through are multiple
More informationUNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test
UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following
More informationcm 2 /second and the height is 10 cm? Please use
Hillary Lehman Writing Assignment Calculus 151 Summer In calculus, there are many different types of problems that may be difficult for students to comprehend. One type of problem that may be difficult,
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 informationSUMMER MATH PACKET students. Entering Geometry-2B
SUMMER MATH PACKET students Entering Geometry-2B The problems in this packet have been selected to help you to review concepts in preparation for your next math class. Please complete the odd problems
More informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More informationMAC Calculus II Spring Homework #6 Some Solutions.
MAC 2312-15931-Calculus II Spring 23 Homework #6 Some Solutions. 1. Find the centroid of the region bounded by the curves y = 2x 2 and y = 1 2x 2. Solution. It is obvious, by inspection, that the centroid
More informationDIFFERENTIATION AND INTEGRATION PART 1. Mr C s IB Standard Notes
DIFFERENTIATION AND INTEGRATION PART 1 Mr C s IB Standard Notes In this PDF you can find the following: 1. Notation 2. Keywords Make sure you read through everything and the try examples for yourself before
More informationMath 180 Written Homework Assignment #10 Due Tuesday, December 2nd at the beginning of your discussion class.
Math 18 Written Homework Assignment #1 Due Tuesday, December 2nd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 18 students, but
More informationThere are some trigonometric identities given on the last page.
MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during
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 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 information(a) The best linear approximation of f at x = 2 is given by the formula. L(x) = f(2) + f (2)(x 2). f(2) = ln(2/2) = ln(1) = 0, f (2) = 1 2.
Math 180 Written Homework Assignment #8 Due Tuesday, November 11th at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180 students,
More informationMATH141: Calculus II Exam #1 review 6/8/2017 Page 1
MATH: Calculus II Eam # review /8/7 Page No review sheet can cover everything that is potentially fair game for an eam, but I tried to hit on all of the topics with these questions, as well as show you
More informationFinal practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90
Final practice, Math 31A - Lec 1, Fall 13 Name and student ID: Question Points Score 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 Total: 9 1. a) 4 points) Find all points x at which the function fx) x 4x + 3 + x
More information4.4: Optimization. Problem 2 Find the radius of a cylindrical container with a volume of 2π m 3 that minimizes the surface area.
4.4: Optimization Problem 1 Suppose you want to maximize a continuous function on a closed interval, but you find that it only has one local extremum on the interval which happens to be a local minimum.
More informationIntegration. Tuesday, December 3, 13
4 Integration 4.3 Riemann Sums and Definite Integrals Objectives n Understand the definition of a Riemann sum. n Evaluate a definite integral using properties of definite integrals. 3 Riemann Sums 4 Riemann
More information18.02 Multivariable Calculus Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. V9. Surface Integrals Surface
More informationLesson 6 Plane Geometry Practice Test Answer Explanations
Lesson 6 Plane Geometry Practice Test Answer Explanations Question 1 One revolution is equal to one circumference: C = r = 6 = 1, which is approximately 37.68 inches. Multiply that by 100 to get 3,768
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 informationAP Calculus AB 2nd Semester Homework List
AP Calculus AB 2nd Semester Homework List Date Assigned: 1/4 DUE Date: 1/6 Title: Typsetting Basic L A TEX and Sigma Notation Write the homework out on paper. Then type the homework on L A TEX. Use this
More informationMULTIVARIABLE CALCULUS
MULTIVARIABLE CALCULUS Summer Assignment Welcome to Multivariable Calculus, Multivariable Calculus is a course commonly taken by second and third year college students. The general concept is to take the
More informationWorksheet #1. A little review.
Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves
More informationGraphs of Polynomial Functions
Graphs of Polynomial Functions By: OpenStaxCollege The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link]. Year 2006 2007 2008 2009 2010 2011 2012 2013
More informationLevel 3 Calculus, 2015
91579 915790 3SUPERVISOR S Level 3 Calculus, 2015 91579 Apply integration methods in solving problems 2.00 p.m. Wednesday 25 November 2015 Credits: Six Achievement Achievement with Merit Achievement with
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More informationMath 221 Exam III (50 minutes) Friday April 19, 2002 Answers
Math Exam III (5 minutes) Friday April 9, Answers I. ( points.) Fill in the boxes as to complete the following statement: A definite integral can be approximated by a Riemann sum. More precisely, if a
More informationMath 142, Final Exam. 12/7/10.
Math 4, Final Exam. /7/0. No notes, calculator, or text. There are 00 points total. Partial credit may be given. Write your full name in the upper right corner of page. Number the pages in the upper right
More informationCalculus II/III Summer Packet
Calculus II/III Summer Packet First of all, have a great summer! Enjoy your time away from school. Come back fired up and ready to learn. I know that I will be ready to have a great year of calculus with
More informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationMath 142, Final Exam, Fall 2006, Solutions
Math 4, Final Exam, Fall 6, Solutions There are problems. Each problem is worth points. SHOW your wor. Mae your wor be coherent and clear. Write in complete sentences whenever this is possible. CIRCLE
More informationMath 1500 Fall 2010 Final Exam Review Solutions
Math 500 Fall 00 Final Eam Review Solutions. Verify that the function f() = 4 + on the interval [, 5] satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
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 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 informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More informationKEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila
KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila February 9, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic
More informationS56 (5.1) Integration.notebook March 09, 2017
Today we will be learning about integration (indefinite integrals) Integration What would you get if you undo the differentiation? Integration is the reverse process of differentiation. It is sometimes
More information4 The Trigonometric Functions
Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater
More informationCalculus II - Fall 2013
Calculus II - Fall Midterm Exam II, November, In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.. Find the area between
More informationMATH 162. Midterm 2 ANSWERS November 18, 2005
MATH 62 Midterm 2 ANSWERS November 8, 2005. (0 points) Does the following integral converge or diverge? To get full credit, you must justify your answer. 3x 2 x 3 + 4x 2 + 2x + 4 dx You may not be able
More informationMa 1c Practical - Solutions to Homework Set 7
Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut
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 informationThings You Should Know Coming Into Calc I
Things You Should Know Coming Into Calc I Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents. Let x and y be positive real numbers, let a and b represent real
More information