B. Examples Set up the integral(s) needed to find the area of the region bounded by
|
|
- Felicia Morton
- 5 years ago
- Views:
Transcription
1 Math 176 Calculus Sec. 6.1: Area Between Curves I. Area between the Curve and the x Axis A. Let f(x) 0 be continuous on [a,b]. The area of the region between the graph of f and the x-axis is A = f ( x) dx. b a Let f(x) 0 be continuous on [a,b]. The area of the region between the graph of f b and the x-axis is A = f ( x) dx a B. Examples Set up the integral(s) needed to find the area of the region bounded by 1. y=x 3, the x-axis and x=1 and x=2. 2. y=x 2 -x-2 and the x-axis.
2 3. y=1-x 2, the x-axis on [0,2]. II. Overview of Area of a Region Between Two Curves With a few modifications the area under a curve represented by a definite integral can be extended to find the area between to curves. Observe the following graphs of f(x) and g(x). Since both graphs lie above the x - axis, we can geometrically interpret the area of the region between the graphs as the area of the region under f minus the area of the region under g.
3 Although we have just shown one case, i.e. when f and g are both positive, this condition is not necessary. All we need in order to evaluate the integrand [f(x) - g(x)] is that f and g both be continuous and g(x) < f(x) on the interval [a,b]. III. Area of a Region Between Two Curves A. Finding the Area Between the Curves Using Vertical Rectangles; Integrating wrt x. 1. If f and g are continuous on [a,b] and g(x) < f(x) for all x in [a, b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ] A = f ( x) g ( x ) dx ; where f(x) is the upper curve & g(x) is the lower curve. a In the next picture, you can see that we simply partition the interval [a,b] and draw our representative rectangles. I've drawn a few rectangles to illustrate the concept. B. Steps to Find the Area Between Two Curves 1. Graph the curves and draw a representative rectangle. This reveals which curve is f (upper curve) and which is g (lower curve). It also helps find the limits of integration if you do not already know them. 2. Find the limits of integration; you may need to find the pts of intersection. 3. Write a formula for f(x) - g(x). Simplify it if you can. 4. Integrate [f(x) - g{x}] from a to b. The number you get is the area. Note: You can use a graphing calculator, Matlab or Mathematica to draw the functions, this will greatly speed up your work. The main reason for doing step 1 is to see how many times the graphs cross (if ever) in the specified domain. It also is to determine which function is dominant in which intervals. If you don't have a quick means to graph the functions, you can set the two functions equal to each other and try to solve for where they are equal. This will give you the intersection points of the two graphs. Once you know these, you can determine which function is dominant in which subintervals. Then just proceed as before. You can use Mathematica's Solve or FindRoot commands to help in finding intersection points.
4 C. Examples 1. Find the area of the region bounded by y=x 2 & y=x Find the area of the region below that is bounded by y=1-x 2, y=(x-1) 2 & y=1.
5 3. Find the area of the region bounded by y=x 3, y=2 -x 2, x=0 & x=2.
6 IV. Area of a Region Between Two Curves Using Horizontal Rectangles; Integrating wrt y. What is really neat is that instead of integrating with respect to x, you can also integrate with respect to y, (your representative rectangles will now be parallel to the x - axis instead of the y - axis). For Example, suppose we had the following graphs: In the first picture, we only need to integrate one time. In the second picture, we would have to integrate twice using different limits of integration and different functions (this is symbolized by the different color rectangles that you would have to construct). A. Finding the Area Between the Curves Using Horizontal Rectangles; Integrating wrt y. 1. If h and k are continuous on [c,d] and k(y) < h(y) for all y in [c,d], then the area of the region bounded by the graphs of h and k and the horizontal lines y=c and y=d is d [ ] A = h( y) k( y) dy ; where h(y) is the right curve & k(y) is the left curve. c 2. Steps to Find the Area Between Two Curves 1. Graph the curves and draw a representative rectangle. This reveals which curve is h (right curve) and which is k (left curve). It also helps find the limits of integration if you do not already know them. 2. Find the limits of integration; you may need to find the pts of intersection. 3. Write a formula for h(y) - k(y). Simplify it if you can. 3. Integrate [h(y) - k(y)] from c to d. The number you get is the area.
7 B. Examples 1. Find the area of the region bounded by x+y 2 =4 & x-y =2. 4. Find the area of the region below that is bounded by y=1-x 2, y=(x-1) 2 & y=1.
Math 1206 Calculus Sec. 5.6: Substitution and Area Between Curves (Part 2) Overview of Area Between Two Curves
Math 1206 Calculus Sec. 5.6: Substitution and Area Between Curves (Part 2) III. Overview of Area Between Two Curves With a few modifications the area under a curve represented by a definite integral can
More informationApplications of Integration. Copyright Cengage Learning. All rights reserved.
Applications of Integration Copyright Cengage Learning. All rights reserved. Area of a Region Between Two Curves Copyright Cengage Learning. All rights reserved. Objectives Find the area of a region between
More informationAP * Calculus Review. Area and Volume
AP * Calculus Review Area and Volume Student Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production of,
More informationIn this chapter, we will investigate what have become the standard applications of the integral:
Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,
More informationAP Calculus AB Unit 2 Assessment
Class: Date: 203-204 AP Calculus AB Unit 2 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. A calculator may NOT be used on this part of the exam.
More informationThe base of a solid is the region in the first quadrant bounded above by the line y = 2, below by
Chapter 7 1) (AB/BC, calculator) The base of a solid is the region in the first quadrant bounded above by the line y =, below by y sin 1 x, and to the right by the line x = 1. For this solid, each cross-section
More informationDouble integrals over a General (Bounded) Region
ouble integrals over a General (Bounded) Region Recall: Suppose z f (x, y) iscontinuousontherectangularregionr [a, b] [c, d]. We define the double integral of f on R by R f (x, y) da lim max x i, y j!0
More informationPolar (BC Only) They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative
Polar (BC Only) Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole
More informationNotice that the height of each rectangle is and the width of each rectangle is.
Math 1410 Worksheet #40: Section 6.3 Name: In some cases, computing the volume of a solid of revolution with cross-sections can be difficult or even impossible. Is there another way to compute volumes
More informationHomework: Study 6.1 # 1, 5, 7, 13, 25, 19; 3, 17, 27, 53
January, 7 Goals:. Remember that the area under a curve is the sum of the areas of an infinite number of rectangles. Understand the approach to finding the area between curves.. Be able to identify the
More informationDirection Fields; Euler s Method
Direction Fields; Euler s Method It frequently happens that we cannot solve first order systems dy (, ) dx = f xy or corresponding initial value problems in terms of formulas. Remarkably, however, this
More informationCalculus III. Math 233 Spring In-term exam April 11th. Suggested solutions
Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total
More informationWorksheet 3.1: Introduction to Double Integrals
Boise State Math 75 (Ultman) Worksheet.: Introduction to ouble Integrals Prerequisites In order to learn the new skills and ideas presented in this worksheet, you must: Be able to integrate functions of
More informationAP Calculus AB Worksheet Areas, Volumes, and Arc Lengths
WorksheetAreasVolumesArcLengths.n 1 AP Calculus AB Worksheet Areas, Volumes, and Arc Lengths Areas To find the area etween the graph of f(x) and the x-axis from x = a to x = we first determine if the function
More information= f (a, b) + (hf x + kf y ) (a,b) +
Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals
More informationFunctions. Copyright Cengage Learning. All rights reserved.
Functions Copyright Cengage Learning. All rights reserved. 2.2 Graphs Of Functions Copyright Cengage Learning. All rights reserved. Objectives Graphing Functions by Plotting Points Graphing Functions with
More information4.7 Approximate Integration
4.7 Approximate Integration Some anti-derivatives are difficult to impossible to find. For example, 1 0 e x2 dx or 1 1 1 + x3 dx We came across this situation back in calculus I when we introduced the
More informationChapter 8: Applications of Definite Integrals
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 8: Applications of Definite Integrals v v Sections: 8.1 Integral as Net Change 8.2 Areas in the Plane v 8.3 Volumes HW Sets Set A (Section 8.1) Pages
More information38. Triple Integration over Rectangular Regions
8. Triple Integration over Rectangular Regions A rectangular solid region S in R can be defined by three compound inequalities, a 1 x a, b 1 y b, c 1 z c, where a 1, a, b 1, b, c 1 and c are constants.
More informationSPM Add Math Form 5 Chapter 3 Integration
SPM Add Math Form Chapter Integration INDEFINITE INTEGRAL CHAPTER : INTEGRATION Integration as the reverse process of differentiation ) y if dy = x. Given that d Integral of ax n x + c = x, where c is
More information2.2 Volumes of Solids of Revolution
2.2 Volumes of Solids of Revolution We know how to find volumes of well-established solids such as a cylinder or rectangular box. What happens when the volume can t be found quite as easily nice or when
More information4 Visualization and. Approximation
4 Visualization and Approximation b A slope field for the differential equation y tan(x + y) tan(x) tan(y). It is not always possible to write down an explicit formula for the solution to a differential
More informationNot for reproduction
x=a GRAPHING CALCULATORS AND COMPUTERS (a, d ) y=d (b, d ) (a, c ) y=c (b, c) (a) _, by _, 4 x=b FIGURE 1 The viewing rectangle a, b by c, d _4 4 In this section we assume that you have access to a graphing
More information3. The domain of a function of 2 or 3 variables is a set of pts in the plane or space respectively.
Math 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.1: Functions of Several Variables I. Functions and Variables A. Def n : Suppose D is a set of n-tuples of real numbers (x 1, x 2,
More informationAP Calculus. Slide 1 / 95. Slide 2 / 95. Slide 3 / 95. Applications of Definite Integrals
Slide 1 / 95 Slide 2 / 95 AP Calculus Applications of Definite Integrals 2015-11-23 www.njctl.org Table of Contents Slide 3 / 95 Particle Movement Area Between Curves Volume: Known Cross Sections Volume:
More informationChanging Variables in Multiple Integrals
Changing Variables in Multiple Integrals 3. Examples and comments; putting in limits. If we write the change of variable formula as (x, y) (8) f(x, y)dx dy = g(u, v) dudv, (u, v) where (x, y) x u x v (9)
More informationSection 6.1 Estimating With Finite Sums
Suppose that a jet takes off, becomes airborne at a velocity of 180 mph and climbs to its cruising altitude. The following table gives the velocity every hour for the first 5 hours, a time during which
More information1.5 Part - 2 Inverse Relations and Inverse Functions
1.5 Part - 2 Inverse Relations and Inverse Functions What happens when we reverse the coordinates of all the ordered pairs in a relation? We obviously get another relation, but does it have any similarities
More informationMath Exam 2a. 1) Take the derivatives of the following. DO NOT SIMPLIFY! 2 c) y = tan(sec2 x) ) b) y= , for x 2.
Math 111 - Exam 2a 1) Take the derivatives of the following. DO NOT SIMPLIFY! a) y = ( + 1 2 x ) (sin(2x) - x- x 1 ) b) y= 2 x + 1 c) y = tan(sec2 x) 2) Find the following derivatives a) Find dy given
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.5 Exponential Functions In this section, we will learn about: Exponential functions and their applications. EXPONENTIAL FUNCTIONS The function f(x) = 2 x is
More informationChapter 5 Accumulating Change: Limits of Sums and the Definite Integral
Chapter 5 Accumulating Change: Limits of Sums and the Definite Integral 5.1 Results of Change and Area Approximations So far, we have used Excel to investigate rates of change. In this chapter we consider
More informationCore Mathematics 1 Graphs of Functions
Regent College Maths Department Core Mathematics 1 Graphs of Functions Graphs of Functions September 2011 C1 Note Graphs of functions; sketching curves defined by simple equations. Here are some curves
More informationProblem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page Mark Sparks 2012
Problem # Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 490 Mark Sparks 01 Finding Anti-derivatives of Polynomial-Type Functions If you had to explain to someone how to find
More informationMATH 19520/51 Class 6
MATH 19520/51 Class 6 Minh-Tam Trinh University of Chicago 2017-10-06 1 Review partial derivatives. 2 Review equations of planes. 3 Review tangent lines in single-variable calculus. 4 Tangent planes to
More information5/27/12. Objectives 7.1. Area of a Region Between Two Curves. Find the area of a region between two curves using integration.
Objectives 7.1 Find the area of a region between two curves using integration. Find the area of a region between intersecting curves using integration. Describe integration as an accumulation process.
More informationAP CALCULUS BC 2013 SCORING GUIDELINES
AP CALCULUS BC 2013 SCORING GUIDELINES Question 4 The figure above shows the graph of f, the derivative of a twice-differentiable function f, on the closed interval 0 x 8. The graph of f has horizontal
More informationWebAssign Lesson 1-2a Area Between Curves (Homework)
WebAssign Lesson 1-2a Area Between Curves (Homework) Current Score : / 30 Due : Thursday, June 26 2014 11:00 AM MDT Jaimos Skriletz Math 175, section 31, Summer 2 2014 Instructor: Jaimos Skriletz 1. /3
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationName: Date: 1. Match the equation with its graph. Page 1
Name: Date: 1. Match the equation with its graph. y 6x A) C) Page 1 D) E) Page . Match the equation with its graph. ( x3) ( y3) A) C) Page 3 D) E) Page 4 3. Match the equation with its graph. ( x ) y 1
More informationChapter 6 Some Applications of the Integral
Chapter 6 Some Applications of the Integral More on Area More on Area Integrating the vertical separation gives Riemann Sums of the form More on Area Example Find the area A of the set shaded in Figure
More informationTopic 6: Calculus Integration Volume of Revolution Paper 2
Topic 6: Calculus Integration Standard Level 6.1 Volume of Revolution Paper 1. Let f(x) = x ln(4 x ), for < x
More informationQUIZ 4 (CHAPTER 17) SOLUTIONS MATH 252 FALL 2008 KUNIYUKI SCORED OUT OF 125 POINTS MULTIPLIED BY % POSSIBLE
QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 5 FALL 8 KUNIYUKI SCORED OUT OF 15 POINTS MULTIPLIED BY.84 15% POSSIBLE 1) Reverse the order of integration, and evaluate the resulting double integral: 16 y dx dy. Give
More informationThe diagram above shows a sketch of the curve C with parametric equations
1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate
More informationThe following information is for reviewing the material since Exam 3:
Outcomes List for Math 121 Calculus I Fall 2010-2011 General Information: The purpose of this Outcomes List is to give you a concrete summary of the material you should know, and the skills you should
More informationMATH 31A HOMEWORK 9 (DUE 12/6) PARTS (A) AND (B) SECTION 5.4. f(x) = x + 1 x 2 + 9, F (7) = 0
FROM ROGAWSKI S CALCULUS (2ND ED.) SECTION 5.4 18.) Express the antiderivative F (x) of f(x) satisfying the given initial condition as an integral. f(x) = x + 1 x 2 + 9, F (7) = 28.) Find G (1), where
More informationArea and Volume. where x right and x left are written in terms of y.
Area and Volume Area between two curves Sketch the region and determine the points of intersection. Draw a small strip either as dx or dy slicing. Use the following templates to set up a definite integral:
More informationAlgebra II. Slide 1 / 181. Slide 2 / 181. Slide 3 / 181. Conic Sections Table of Contents
Slide 1 / 181 Algebra II Slide 2 / 181 Conic Sections 2015-04-21 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 181 Review of Midpoint and Distance Formulas Introduction
More informationMath 113 Calculus III Final Exam Practice Problems Spring 2003
Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross
More informationRational Numbers: Graphing: The Coordinate Plane
Rational Numbers: Graphing: The Coordinate Plane A special kind of plane used in mathematics is the coordinate plane, sometimes called the Cartesian plane after its inventor, René Descartes. It is one
More informationAP Calculus BC. Find a formula for the area. B. The cross sections are squares with bases in the xy -plane.
AP Calculus BC Find a formula for the area Homework Problems Section 7. Ax of the cross sections of the solid that are perpendicular to the x -axis. 1. The solid lies between the planes perpendicular to
More informationx 6 + λ 2 x 6 = for the curve y = 1 2 x3 gives f(1, 1 2 ) = λ actually has another solution besides λ = 1 2 = However, the equation λ
Math 0 Prelim I Solutions Spring 010 1. Let f(x, y) = x3 y for (x, y) (0, 0). x 6 + y (4 pts) (a) Show that the cubic curves y = x 3 are level curves of the function f. Solution. Substituting y = x 3 in
More informationMath 113 Exam 1 Practice
Math Exam Practice January 6, 00 Exam will cover sections 6.-6.5 and 7.-7.5 This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second
More informationUniversity of California, Berkeley
University of California, Berkeley FINAL EXAMINATION, Fall 2012 DURATION: 3 hours Department of Mathematics MATH 53 Multivariable Calculus Examiner: Sean Fitzpatrick Total: 100 points Family Name: Given
More informationExam 1 Review. MATH Intuitive Calculus Fall Name:. Show your reasoning. Use standard notation correctly.
MATH 11012 Intuitive Calculus Fall 2012 Name:. Exam 1 Review Show your reasoning. Use standard notation correctly. 1. Consider the function f depicted below. y 1 1 x (a) Find each of the following (or
More informationTHE UNIVERSITY OF AKRON Mathematics and Computer Science Article: Applications to Definite Integration
THE UNIVERSITY OF AKRON Mathematics and Computer Science Article: Applications to Definite Integration calculus menu Directory Table of Contents Begin tutorial on Integration Index Copyright c 1995 1998
More informationUpdated: March 31, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University
Updated: March 3, 26 Calculus III Section 5.6 Math 232 Calculus III Brian Veitch Fall 25 Northern Illinois University 5.6 Triple Integrals In order to build up to a triple integral let s start back at
More informationVolumes of Solids of Revolution
Volumes of Solids of Revolution Farid Aliniaeifard York University http://math.yorku.ca/ faridanf April 27, 2016 Overview What is a solid of revolution? Method of Rings or Method of Disks Method of Cylindrical
More informationSolved Examples. Parabola with vertex as origin and symmetrical about x-axis. We will find the area above the x-axis and double the area.
Solved Examples Example 1: Find the area common to the curves x 2 + y 2 = 4x and y 2 = x. x 2 + y 2 = 4x (i) (x 2) 2 + y 2 = 4 This is a circle with centre at (2, 0) and radius 2. y = (4x-x 2 ) y 2 = x
More informationSection 2.1 Graphs. The Coordinate Plane
Section 2.1 Graphs The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of numbers to form
More informationheight VUD x = x 1 + x x N N 2 + (x 2 x) 2 + (x N x) 2. N
Math 3: CSM Tutorial: Probability, Statistics, and Navels Fall 2 In this worksheet, we look at navel ratios, means, standard deviations, relative frequency density histograms, and probability density functions.
More information0.7 Graphing Features: Value (Eval), Zoom, Trace, Maximum/Minimum, Intersect
0.7 Graphing Features: Value (Eval), Zoom, Trace, Maximum/Minimum, Intersect Value (TI-83 and TI-89), Eval (TI-86) The Value or Eval feature allows us to enter a specific x coordinate and the cursor moves
More informationB.Stat / B.Math. Entrance Examination 2017
B.Stat / B.Math. Entrance Examination 017 BOOKLET NO. TEST CODE : UGA Forenoon Questions : 0 Time : hours Write your Name, Registration Number, Test Centre, Test Code and the Number of this Booklet in
More informationAP CALCULUS BC PACKET 2 FOR UNIT 4 SECTIONS 6.1 TO 6.3 PREWORK FOR UNIT 4 PT 2 HEIGHT UNDER A CURVE
AP CALCULUS BC PACKET FOR UNIT 4 SECTIONS 6. TO 6.3 PREWORK FOR UNIT 4 PT HEIGHT UNDER A CURVE Find an expression for the height of an vertical segment that can be drawn into the shaded region... = x =
More informationCoordinate Transformations in Advanced Calculus
Coordinate Transformations in Advanced Calculus by Sacha Nandlall T.A. for MATH 264, McGill University Email: sacha.nandlall@mail.mcgill.ca Website: http://www.resanova.com/teaching/calculus/ Fall 2006,
More informationMath 126 Final Examination Autumn CHECK that your exam contains 9 problems on 10 pages.
Math 126 Final Examination Autumn 2016 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name CHECK that your exam contains 9 problems on 10 pages. This exam is closed book. You
More informationSharp EL-9900 Graphing Calculator
Sharp EL-9900 Graphing Calculator Basic Keyboard Activities General Mathematics Algebra Programming Advanced Keyboard Activities Algebra Calculus Statistics Trigonometry Programming Sharp EL-9900 Graphing
More informationMATH 261 EXAM III PRACTICE PROBLEMS
MATH 6 EXAM III PRACTICE PROBLEMS These practice problems are pulled from actual midterms in previous semesters. Exam 3 typically has 5 (not 6!) problems on it, with no more than one problem of any given
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS In this section, we assume that you have access to a graphing calculator or a computer with graphing software. FUNCTIONS AND MODELS 1.4 Graphing Calculators
More informationWe imagine the egg being the three dimensional solid defined by rotating this ellipse around the x-axis:
CHAPTER 6. INTEGRAL APPLICATIONS 7 Example. Imagine we want to find the volume of hard boiled egg. We could put the egg in a measuring cup and measure how much water it displaces. But we suppose we want
More informationMATH 104 Sample problems for first exam - Fall MATH 104 First Midterm Exam - Fall (d) 256 3
MATH 14 Sample problems for first exam - Fall 1 MATH 14 First Midterm Exam - Fall 1. Find the area between the graphs of y = 9 x and y = x + 1. (a) 4 (b) (c) (d) 5 (e) 4 (f) 81. A solid has as its base
More informationEuler s Method for Approximating Solution Curves
Euler s Method for Approximating Solution Curves As you may have begun to suspect at this point, time constraints will allow us to learn only a few of the many known methods for solving differential equations.
More informationNAME: Section # SSN: X X X X
Math 155 FINAL EXAM A May 5, 2003 NAME: Section # SSN: X X X X Question Grade 1 5 (out of 25) 6 10 (out of 25) 11 (out of 20) 12 (out of 20) 13 (out of 10) 14 (out of 10) 15 (out of 16) 16 (out of 24)
More informationMA 114 Worksheet #17: Average value of a function
Spring 2019 MA 114 Worksheet 17 Thursday, 7 March 2019 MA 114 Worksheet #17: Average value of a function 1. Write down the equation for the average value of an integrable function f(x) on [a, b]. 2. Find
More informationLab 2B Parametrizing Surfaces Math 2374 University of Minnesota Questions to:
Lab_B.nb Lab B Parametrizing Surfaces Math 37 University of Minnesota http://www.math.umn.edu/math37 Questions to: rogness@math.umn.edu Introduction As in last week s lab, there is no calculus in this
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. CV. Changing Variables in
More informationLecture 15. Lecturer: Prof. Sergei Fedotov Calculus and Vectors. Length of a Curve and Parametric Equations
Lecture 15 Lecturer: Prof. Sergei Fedotov 10131 - Calculus and Vectors Length of a Curve and Parametric Equations Sergei Fedotov (University of Manchester) MATH10131 2011 1 / 5 Lecture 15 1 Length of a
More informationProperties of a Function s Graph
Section 3.2 Properties of a Function s Graph Objective 1: Determining the Intercepts of a Function An intercept of a function is a point on the graph of a function where the graph either crosses or touches
More informationUnit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1
Algebra I Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola Name Period Date Day #1 There are some important features about the graphs of quadratic functions we are going to explore over the
More informationform are graphed in Cartesian coordinates, and are graphed in Cartesian coordinates.
Plot 3D Introduction Plot 3D graphs objects in three dimensions. It has five basic modes: 1. Cartesian mode, where surfaces defined by equations of the form are graphed in Cartesian coordinates, 2. cylindrical
More informationAP Calculus. Areas and Volumes. Student Handout
AP Calculus Areas and Volumes Student Handout 016-017 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss Copyright
More information1. (12 points) Find an equation for the line tangent to the graph of f(x) = xe 2x+4 at the point (2, f(2)).
April 13, 2011 Name The problems count as marked The total number of points available is 159 Throughout this test, show your work Use calculus to work the problems Calculator solutions which circumvent
More information1. Show that the rectangle of maximum area that has a given perimeter p is a square.
Constrained Optimization - Examples - 1 Unit #23 : Goals: Lagrange Multipliers To study constrained optimization; that is, the maximizing or minimizing of a function subject to a constraint (or side condition).
More informationMath 2260 Exam #1 Practice Problem Solutions
Math 6 Exam # Practice Problem Solutions. What is the area bounded by the curves y x and y x + 7? Answer: As we can see in the figure, the line y x + 7 lies above the parabola y x in the region we care
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13
MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus
More informationc x y f() f (x) Determine the Determine the Approximate c : Replacin on the AP exam: under-approximation
Tangent Lines and Linear Approximations Students should be able to: Determine the slope of tangent line to a curve at a point Determine the equations of tangent lines and normal lines Approximate a value
More informationREVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections
REVIEW I MATH 254 Calculus IV Exam I (Friday, April 29 will cover sections 14.1-8. 1. Functions of multivariables The definition of multivariable functions is similar to that of functions of one variable.
More informationMATH 122 FINAL EXAM WINTER March 15, 2011
MATH 1 FINAL EXAM WINTER 010-011 March 15, 011 NAME: SECTION: ONLY THE CORRECT ANSWER AND ALL WORK USED TO REACH IT WILL EARN FULL CREDIT. Simplify all answers as much as possible unless explicitly stated
More informationApproximate Solutions to Differential Equations Slope Fields (graphical) and Euler s Method (numeric) by Dave Slomer
Approximate Solutions to Differential Equations Slope Fields (graphical) and Euler s Method (numeric) by Dave Slomer Leonhard Euler was a great Swiss mathematician. The base of the natural logarithm function,
More informationMath 115 Second Midterm March 25, 2010
Math 115 Second Midterm March 25, 2010 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover. There are 8 problems. Note that the
More informationPart I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each.
Math 106/108 Final Exam Page 1 Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each. 1. Factor completely. Do not solve. a) 2x
More informationMATH 19520/51 Class 15
MATH 19520/51 Class 15 Minh-Tam Trinh University of Chicago 2017-11-01 1 Change of variables in two dimensions. 2 Double integrals via change of variables. Change of Variables Slogan: An n-variable substitution
More informationI IS II. = 2y"\ V= n{ay 2 l 3 -\y 2 )dy. Jo n [fy 5 ' 3 1
r Exercises 5.2 Figure 530 (a) EXAMPLE'S The region in the first quadrant bounded by the graphs of y = i* and y = 2x is revolved about the y-axis. Find the volume of the resulting solid. SOLUTON The region
More information4.2 and 4.6 filled in notes.notebook. December 08, Integration. Copyright Cengage Learning. All rights reserved.
4 Integration Copyright Cengage Learning. All rights reserved. 1 4.2 Area Copyright Cengage Learning. All rights reserved. 2 Objectives Use sigma notation to write and evaluate a sum. Understand the concept
More informationMath 265 Exam 3 Solutions
C Roettger, Fall 16 Math 265 Exam 3 Solutions Problem 1 Let D be the region inside the circle r 5 sin θ but outside the cardioid r 2 + sin θ. Find the area of D. Note that r and θ denote polar coordinates.
More information----- o Implicit Differentiation ID: A. dy r.---; d 2 Y 2. If- = '" 1-y- then - = dx 'dx 2. a c. -1 d. -2 e.
Name: Class: Date: ----- ID: A Implicit Differentiation Multiple Choice Identify the choice that best completes the statement or answers the question.. The slope of the line tangent to the curve y + (xy
More informationINTRODUCTION TO LINE INTEGRALS
INTRODUTION TO LINE INTEGRALS PROF. MIHAEL VANVALKENBURGH Last week we discussed triple integrals. This week we will study a new topic of great importance in mathematics and physics: line integrals. 1.
More informationMath 253, Section 102, Fall 2006 Practice Final Solutions
Math 253, Section 102, Fall 2006 Practice Final Solutions 1 2 1. Determine whether the two lines L 1 and L 2 described below intersect. If yes, find the point of intersection. If not, say whether they
More informationQUADRATIC AND CUBIC GRAPHS
NAME SCHOOL INDEX NUMBER DATE QUADRATIC AND CUBIC GRAPHS KCSE 1989 2012 Form 3 Mathematics Working Space 1. 1989 Q22 P1 (a) Using the grid provided below draw the graph of y = -2x 2 + x + 8 for values
More informationCCNY Math Review Chapter 2: Functions
CCN Math Review Chapter : Functions Section.1: Functions.1.1: How functions are used.1.: Methods for defining functions.1.3: The graph of a function.1.: Domain and range.1.5: Relations, functions, and
More informationt dt ds Then, in the last class, we showed that F(s) = <2s/3, 1 2s/3, s/3> is arclength parametrization. Therefore,
13.4. Curvature Curvature Let F(t) be a vector values function. We say it is regular if F (t)=0 Let F(t) be a vector valued function which is arclength parametrized, which means F t 1 for all t. Then,
More information