An Introduction to Double Integrals Math Insight
|
|
- Kellie Ford
- 5 years ago
- Views:
Transcription
1 An Introduction to Double Integrals Math Insight Suppose that you knew the hair density at each point on your head and you wanted to calculate the total number of hairs on your head. In other words, let (x,y) be a point that is x millimeters to the right and y millimeters above some reference point, say your nose. We are assuming that you already know the function f(x,y) that gives hair density in hairs per square millimeter at point (x,y). The following might be a method to use your knowledge of f(x,y) to estimate the number of hairs on your body. 1. Cut and flatten your skin so it lies in a plane. (Although this has nothing to do with double integrals, brain mappers and cartographers face similar problems. To map the brain or the surface of the earth, one looks for ways to flatten these surfaces into a plane.) 2. Divide your skin into small rectangles of width Δx and height Δy. 3. Label each rectangle according to row i and column j. For rectangle ij, pick some point in the rectangle, and call it (xij,yij). Since you know your hair density function, you can look up the hair density of that point. It is simply f(xij,yij). Label each rectangle by the number f(xij,yij). 4. If the hair density were constant in each rectangle, the number of hairs in rectangle ij would be f(xij,yij) times the area of the rectangle. The area of the rectangle is simply its width (Δx) times its height (Δy), i.e., the area is ΔxΔy. Hence, the number of hairs in the rectangle is approximately f(xij,yij)δxδy. Saylor URL: Page 1 of 5
2 5. To estimate the total number of hairs on your head, you can add up the (approximate) number of hairs in each rectangle. Using the above result, your estimate for the total number of hairs is where the sum is over all rectangles. If in the above picture, each rectangle were 75 millimeters wide and 65 millimeters high, then the resulting estimate of the total number of hairs would be ( ) 75 65=609, The above result is only a rough estimate because it assumed that the hair density was constant over each rectangle. You may also have noticed that additional errors are introduced around the edges of your skin, where some rectangles are only partly filled with skin. You can increase your accuracy by decreasing the size of each rectangle (i.e., decreasing Δx and Δy). Of course, to cover your whole head, you'll have to increase the number of rectangles as you decrease their size. 7. To be really accurate, you should let the size of the rectangles go to zero (and the number of rectangles go to infinity). In other words, you should take the limit where Δx 0 and Δy 0. As long as f(x,y) is a continuous function, this procedure will converge to a single number, which would be the actual number of hairs on your head. 8. We can put this into math terms. When we took the limit as Δx 0 and Δy 0, we ended up with the definite integral of the function f over your head. If we denote by D the region of the plane that your skin occupied, then we write this integral as We refer to this integral as the double integral of f over D. The sums of step 5 are the Riemann sums that approximate the integral. The integral is the limit of the Riemann sums as the size of the rectangles goes to zero. This is exactly the way you defined the integral in one-variable calculus. Saylor URL: Page 2 of 5
3 You can read how we can interpret the double integral as volume 1 underneath a surface, just like you could interpret the regular one-variable integral as area under a curve. In this case, we can also visualize the Riemann sum defining the integral as the volume of many boxes, as illustrated in the below applet. (More details on this volume interpretation and this applet can be viewed on this page 2.) Double integral Riemann sum. The volume of the small boxes illustrates a Riemann sum approximating the volume under the graph of z=f(x,y) over the region D, i.e., the double integral DfdA for f(x,y)=cos2x+sin2y and D defined by 0 x 2 and 0 y 1. The volume of the boxes is where x i is the midpoint of the ith interval along the x-axis and y j is the midpoint of the jth interval along the y-axis. The purple line of the cyan slider shows the volume estimated by the volume of the boxes, and the blue line of the cyan slider shows the actual volume underneath the surface. As Δx and Δy approach zero, the purple line approaches the blue line, illustrating how the estimated volume approaches the actual volume. I don't have good examples (other than the above hair-counting example) on computing double integrals this way, as this is not how we typically compute Saylor URL: Page 3 of 5
4 them. Instead, this page is about how we define a double integral. We have better ways to compute double integrals (that is, unless you are a computer, in which case chopping up the domain in pieces and computing a sum as an approximation to an integral works pretty well). You can also read examples of computing double integrals 3 using the method in which those of us who are not computers typically use, which is something called an iterated integral 4. Saylor URL: Page 4 of 5
5 Notes and Links: Saylor URL: Page 5 of 5
If ( ) is approximated by a left sum using three inscribed rectangles of equal width on the x-axis, then the approximation is
More Integration Page 1 Directions: Solve the following problems using the available space for scratchwork. Indicate your answers on the front page. Do not spend too much time on any one problem. Note:
More informationCHAPTER 8: INTEGRALS 8.1 REVIEW: APPROXIMATING INTEGRALS WITH RIEMANN SUMS IN 2-D
CHAPTER 8: INTEGRALS 8.1 REVIEW: APPROXIMATING INTEGRALS WITH RIEMANN SUMS IN 2-D In two dimensions we have previously used Riemann sums to approximate ( ) following steps: with the 1. Divide the region
More informationAn Introduction to the Directional Derivative and the Gradient Math Insight
An Introduction to the Directional Derivative and the Gradient Math Insight The directional derivative Let the function f(x,y) be the height of a mountain range at each point x=(x,y). If you stand at some
More informationUnit #13 : Integration to Find Areas and Volumes, Volumes of Revolution
Unit #13 : Integration to Find Areas and Volumes, Volumes of Revolution Goals: Beabletoapplyaslicingapproachtoconstructintegralsforareasandvolumes. Be able to visualize surfaces generated by rotating functions
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 informationMath 11 Fall 2016 Section 1 Monday, October 17, 2016
Math 11 Fall 16 Section 1 Monday, October 17, 16 First, some important points from the last class: f(x, y, z) dv, the integral (with respect to volume) of f over the three-dimensional region, is a triple
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 informationR f da (where da denotes the differential of area dxdy (or dydx)
Math 28H Topics for the second exam (Technically, everything covered on the first exam, plus) Constrained Optimization: Lagrange Multipliers Most optimization problems that arise naturally are not unconstrained;
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 informationRectangle Sums
Rectangle Sums --208 You can approximate the area under a curve using rectangles. To do this, divide the base interval into pieces subintervals). Then on each subinterval, build a rectangle that goes up
More informationB. Examples Set up the integral(s) needed to find the area of the region bounded by
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)
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 information2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0.
Midterm 3 Review Short Answer 2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0. 3. Compute the Riemann sum for the double integral where for the given grid
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 informationMath 205 Test 3 Grading Guidelines Problem 1 Part a: 1 point for figuring out r, 2 points for setting up the equation P = ln 2 P and 1 point for the initial condition. Part b: All or nothing. This is really
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 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 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 informationThe Fundamental Theorem of Calculus Using the Rule of Three
The Fundamental Theorem of Calculus Using the Rule of Three A. Approimations with Riemann sums. The area under a curve can be approimated through the use of Riemann (or rectangular) sums: n Area f ( k
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 informationMath 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 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 informationMatrix-Vector Multiplication by MapReduce. From Rajaraman / Ullman- Ch.2 Part 1
Matrix-Vector Multiplication by MapReduce From Rajaraman / Ullman- Ch.2 Part 1 Google implementation of MapReduce created to execute very large matrix-vector multiplications When ranking of Web pages that
More informationThe listing says y(1) = How did the computer know this?
18.03 Class 2, Feb 3, 2010 Numerical Methods [1] How do you know the value of e? [2] Euler's method [3] Sources of error [4] Higher order methods [1] The study of differential equations has three parts:.
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 informationTriple Integrals. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Triple Integrals
Triple Integrals MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 211 Riemann Sum Approach Suppose we wish to integrate w f (x, y, z), a continuous function, on the box-shaped region
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 informationExploring Fractals through Geometry and Algebra. Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss
Exploring Fractals through Geometry and Algebra Kelly Deckelman Ben Eggleston Laura Mckenzie Patricia Parker-Davis Deanna Voss Learning Objective and skills practiced Students will: Learn the three criteria
More informationGrade 6 Math Circles. Spatial and Visual Thinking
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles October 31/November 1, 2017 Spatial and Visual Thinking Centre for Education in Mathematics and Computing One very important
More informationMassachusetts Institute of Technology Department of Computer Science and Electrical Engineering 6.801/6.866 Machine Vision QUIZ II
Massachusetts Institute of Technology Department of Computer Science and Electrical Engineering 6.801/6.866 Machine Vision QUIZ II Handed out: 001 Nov. 30th Due on: 001 Dec. 10th Problem 1: (a (b Interior
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 informationCirculation density and an argument for Stokes Theorem
MATH 221A Multivariate Calculus pring 2004 Circulation density and an argument for tokes Theorem Version 1 (May 13, 2004) There are three sections in these notes. The goal for the first section is to see
More informationFigure 1: Two polygonal loops
Math 1410: The Polygonal Jordan Curve Theorem: The purpose of these notes is to prove the polygonal Jordan Curve Theorem, which says that the complement of an embedded polygonal loop in the plane has exactly
More informationApplications of Triple Integrals
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 informationMIDTERM. Section: Signature:
MIDTERM Math 32B 8/8/2 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. NO CALCULATORS! Show all work, clearly
More informationJohn Perry. Spring 2016
MAT 305: Repeating a task on a set (or list, or tuple, or...) University of Southern Mississippi Spring 2016 Outline 1 2 3 4 5 6 7 Outline 1 2 3 4 5 6 7 Differential Equations What is y in terms of x?
More information3.7, Graphing Calculator Fun. Sketch a full picture of the following graphs. Draw axes and indicate the x and y max and min.
3.7, 3.9 1 Graphing Calculator Fun Sketch a full picture of the following graphs. Draw axes and indicate the x and y max and min. y = x 2 32x +240 y = -2sin(x) +1 [-2π, 2π] Solve the following 3x 2 4x
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 information(Refer Slide Time: 02:59)
Numerical Methods and Programming P. B. Sunil Kumar Department of Physics Indian Institute of Technology, Madras Lecture - 7 Error propagation and stability Last class we discussed about the representation
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 informationAnnouncements. Topics: To Do:
Announcements Topics: - Systems of DEs (8.5) - The Phase Plane (8.6) - Solutions in the Phase Plane (8.7) In the Functions of Several Variables module: - Section 1: Introduction to Functions of Several
More informationAP Computer Science Unit 3. Programs
AP Computer Science Unit 3. Programs For most of these programs I m asking that you to limit what you print to the screen. This will help me in quickly running some tests on your code once you submit them
More informationUNC Charlotte 2010 Comprehensive
00 Comprehensive March 8, 00. A cubic equation x 4x x + a = 0 has three roots, x, x, x. If x = x + x, what is a? (A) 4 (B) 8 (C) 0 (D) (E) 6. For which value of a is the polynomial P (x) = x 000 +ax+9
More informationTable 3: Midpoint estimate
The function y gx.x x 1 is shown in Figure 1. Six midpoint rectangles have been drawn between the function and the x-axis over,1 ; the areas of these six rectangles are shown in Table 1. Figure 1 Let s
More information6-1 THE STANDARD NORMAL DISTRIBUTION
6-1 THE STANDARD NORMAL DISTRIBUTION The major focus of this chapter is the concept of a normal probability distribution, but we begin with a uniform distribution so that we can see the following two very
More informationBasic and Intermediate Math Vocabulary Spring 2017 Semester
Digit A symbol for a number (1-9) Whole Number A number without fractions or decimals. Place Value The value of a digit that depends on the position in the number. Even number A natural number that is
More informationLab 4. Recall, from last lab the commands Table[], ListPlot[], DiscretePlot[], and Limit[]. Go ahead and review them now you'll be using them soon.
Calc II Page 1 Lab 4 Wednesday, February 19, 2014 5:01 PM Recall, from last lab the commands Table[], ListPlot[], DiscretePlot[], and Limit[]. Go ahead and review them now you'll be using them soon. Use
More informationPAF Chapter Prep Section Mathematics Class 6 Worksheets for Intervention Classes
The City School PAF Chapter Prep Section Mathematics Class 6 Worksheets for Intervention Classes Topic: Percentage Q1. Convert it into fractions and its lowest term: a) 25% b) 75% c) 37% Q2. Convert the
More informationThere we are; that's got the 3D screen and mouse sorted out.
Introduction to 3D To all intents and purposes, the world we live in is three dimensional. Therefore, if we want to construct a realistic computer model of it, the model should be three dimensional as
More informationMath 414 Lecture 30. The greedy algorithm provides the initial transportation matrix.
Math Lecture The greedy algorithm provides the initial transportation matrix. matrix P P Demand W ª «2 ª2 «W ª «W ª «ª «ª «Supply The circled x ij s are the initial basic variables. Erase all other values
More informationGraphing on Excel. Open Excel (2013). The first screen you will see looks like this (it varies slightly, depending on the version):
Graphing on Excel Open Excel (2013). The first screen you will see looks like this (it varies slightly, depending on the version): The first step is to organize your data in columns. Suppose you obtain
More informationBulgarian Math Olympiads with a Challenge Twist
Bulgarian Math Olympiads with a Challenge Twist by Zvezdelina Stankova Berkeley Math Circle Beginners Group September 0, 03 Tasks throughout this session. Harder versions of problems from last time appear
More informationChapter 4: Linear Relations
Chapter 4: Linear Relations How many people can sit around 1 table? If you put two tables together, how many will the new arrangement seat? What if there are 10 tables? What if there are 378 tables in
More informationCALCULUS LABORATORY ACTIVITY: Numerical Integration, Part 1
CALCULUS LABORATORY ACTIVITY: Numerical Integration, Part 1 Required tasks: Tabulate values, create sums Suggested Technology: Goals Spreadsheet: Microsoft Excel or Google Docs Spreadsheet Maple or Wolfram
More informationMathematics 350 Section 6.3 Introduction to Fractals
Mathematics 350 Section 6.3 Introduction to Fractals A fractal is generally "a rough or fragmented geometric shape that is self-similar, which means it can be split into parts, each of which is (at least
More informationDouble Integration: Non-Rectangular Domains
Double Integration: Non-Rectangular Domains Thomas Banchoff and Associates June 18, 2003 1 Introduction In calculus of one variable, all domains are intervals which are subsets of the line. In calculus
More informationChapter 3A Rectangular Coordinate System
Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 1 Chapter 3A Rectangular Coordinate System A is any set of ordered pairs of real numbers. The of the relation is the set of all first elements
More informationax + by = 0. x = c. y = d.
Review of Lines: Section.: Linear Inequalities in Two Variables The equation of a line is given by: ax + by = c. for some given numbers a, b and c. For example x + y = 6 gives the equation of a line. A
More informationS206E Lecture 17, 5/1/2018, Rhino & Grasshopper, Tower modeling
S206E057 -- Lecture 17, 5/1/2018, Rhino & Grasshopper, Tower modeling Copyright 2018, Chiu-Shui Chan. All Rights Reserved. Concept of Morph in Rhino and Grasshopper: S206E057 Spring 2018 Morphing is a
More informationMeasures of Dispersion
Lesson 7.6 Objectives Find the variance of a set of data. Calculate standard deviation for a set of data. Read data from a normal curve. Estimate the area under a curve. Variance Measures of Dispersion
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 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 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 informationDirect Variations DIRECT AND INVERSE VARIATIONS 19. Name
DIRECT AND INVERSE VARIATIONS 19 Direct Variations Name Of the many relationships that two variables can have, one category is called a direct variation. Use the description and example of direct variation
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 information4. Write sets of directions for how to check for direct variation. How to check for direct variation by analyzing the graph :
Name Direct Variations There are many relationships that two variables can have. One of these relationships is called a direct variation. Use the description and example of direct variation to help you
More informationMath 250A (Fall 2009) - Lab I: Estimate Integrals Numerically with Matlab. Due Date: Monday, September 21, INSTRUCTIONS
Math 250A (Fall 2009) - Lab I: Estimate Integrals Numerically with Matlab Due Date: Monday, September 21, 2009 4:30 PM 1. INSTRUCTIONS The primary purpose of this lab is to understand how go about numerically
More informationMath 6, Unit 8 Notes: Geometric Relationships
Math 6, Unit 8 Notes: Geometric Relationships Points, Lines and Planes; Line Segments and Rays As we begin any new topic, we have to familiarize ourselves with the language and notation to be successful.
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationOptimization in One Variable Using Solver
Chapter 11 Optimization in One Variable Using Solver This chapter will illustrate the use of an Excel tool called Solver to solve optimization problems from calculus. To check that your installation of
More informationThis is a good time to refresh your memory on double-integration. We will be using this skill in the upcoming lectures.
Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 1: Sections 5-1.1 to 5-1.4 For both discrete and continuous random variables we will discuss the following... Joint Distributions (for two or more r.v. s)
More informationUSING THE DEFINITE INTEGRAL
Print this page Chapter Eight USING THE DEFINITE INTEGRAL 8.1 AREAS AND VOLUMES In Chapter 5, we calculated areas under graphs using definite integrals. We obtained the integral by slicing up the region,
More informationMicroscopic Measurement
Microscopic Measurement Estimating Specimen Size : The area of the slide that you see when you look through a microscope is called the " field of view ". If you know the diameter of your field of view,
More information2.3 Algorithms Using Map-Reduce
28 CHAPTER 2. MAP-REDUCE AND THE NEW SOFTWARE STACK one becomes available. The Master must also inform each Reduce task that the location of its input from that Map task has changed. Dealing with a failure
More informationUnit 3: Triangles and Polygons
Unit 3: Triangles and Polygons Background for Standard G.CO.9: Prove theorems about triangles. Objective: By the end of class, I should Example 1: Trapezoid on the coordinate plane below has the following
More informationCalculus I Review Handout 1.3 Introduction to Calculus - Limits. by Kevin M. Chevalier
Calculus I Review Handout 1.3 Introduction to Calculus - Limits by Kevin M. Chevalier We are now going to dive into Calculus I as we take a look at the it process. While precalculus covered more static
More informationIntroduction to Homogeneous coordinates
Last class we considered smooth translations and rotations of the camera coordinate system and the resulting motions of points in the image projection plane. These two transformations were expressed mathematically
More informationDouble Integrals, Iterated Integrals, Cross-sections
Chapter 14 Multiple Integrals 1 ouble Integrals, Iterated Integrals, Cross-sections 2 ouble Integrals over more general regions, efinition, Evaluation of ouble Integrals, Properties of ouble Integrals
More information12.7 Tangent Planes and Normal Lines
.7 Tangent Planes and Normal Lines Tangent Plane and Normal Line to a Surface Suppose we have a surface S generated by z f(x,y). We can represent it as f(x,y)-z 0 or F(x,y,z) 0 if we wish. Hence we can
More information13.1 2/20/2018. Conic Sections. Conic Sections: Parabolas and Circles
13 Conic Sections 13.1 Conic Sections: Parabolas and Circles 13.2 Conic Sections: Ellipses 13.3 Conic Sections: Hyperbolas 13.4 Nonlinear Systems of Equations 13.1 Conic Sections: Parabolas and Circles
More informationCYLINDRICAL COORDINATES
CHAPTER 11: CYLINDRICAL COORDINATES 11.1 DEFINITION OF CYLINDRICAL COORDINATES A location in 3-space can be defined with (r, θ, z) where (r, θ) is a location in the xy plane defined in polar coordinates
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
Zeros&asymptotes Example 1 In an early version of this activity I began with a sequence of simple examples (parabolas and cubics) working gradually up to the main idea. But now I think the best strategy
More informationUNC Charlotte 2010 Comprehensive
00 Comprehensive with solutions March 8, 00. A cubic equation x 4x x + a = 0 has three roots, x, x, x. If x = x + x, what is a? (A) 4 (B) 8 (C) 0 (D) (E) 6 Solution: C. Because the coefficient of x is
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More information1. The Normal Distribution, continued
Math 1125-Introductory Statistics Lecture 16 10/9/06 1. The Normal Distribution, continued Recall that the standard normal distribution is symmetric about z = 0, so the area to the right of zero is 0.5000.
More informationReading. Parametric surfaces. Surfaces of revolution. Mathematical surface representations. Required:
Reading Required: Angel readings for Parametric Curves lecture, with emphasis on 11.1.2, 11.1.3, 11.1.5, 11.6.2, 11.7.3, 11.9.4. Parametric surfaces Optional Bartels, Beatty, and Barsky. An Introduction
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 informationCREATING CALCULUS DEMOS WITH GEOGEBRA 4
CREATING CALCULUS DEMOS WITH GEOGEBRA 4 Barbara K. D Ambrosia Carl R. Spitznagel John Carroll University Department of Mathematics and Computer Science Cleveland, OH 44118 bdambrosia@jcu.edu spitz@jcu.edu
More informationCS 4349 Lecture September 13th, 2017
CS 4349 Lecture September 13th, 2017 Main topics for #lecture include #dynamic_programming, #Fibonacci_numbers, and #rod_cutting. Prelude Homework 2 due today in class. Homework 3 released, due next Wednesday
More informationGeoGebra. 10 Lessons. maths.com. Gerrit Stols. For more info and downloads go to:
GeoGebra in 10 Lessons For more info and downloads go to: http://school maths.com Gerrit Stols Acknowledgements Download GeoGebra from http://www.geogebra.org GeoGebra is dynamic mathematics open source
More informationIntegration. Example Find x 3 dx.
Integration A function F is called an antiderivative of the function f if F (x)=f(x). The set of all antiderivatives of f is called the indefinite integral of f with respect to x and is denoted by f(x)dx.
More informationTeaching Math thru Big Ideas Focusing on Differentiation. Marian Small April 2017 San Diego, CA
Teaching Math thru Big Ideas Focusing on Differentiation Marian Small April 2017 San Diego, CA Our focus today Differentiation by focusing on big ideas Formative assessment/feedback Planning lessons/units
More informationWinter 2012 Math 255 Section 006. Problem Set 7
Problem Set 7 1 a) Carry out the partials with respect to t and x, substitute and check b) Use separation of varibles, i.e. write as dx/x 2 = dt, integrate both sides and observe that the solution also
More informationChapter 5. Transforming Shapes
Chapter 5 Transforming Shapes It is difficult to walk through daily life without being able to see geometric transformations in your surroundings. Notice how the leaves of plants, for example, are almost
More informationSection 1: Section 2: Section 3: Section 4:
Announcements Topics: In the Functions of Several Variables module: - Section 1: Introduction to Functions of Several Variables (Basic Definitions and Notation) - Section 2: Graphs, Level Curves + Contour
More informationLearning Objectives. Continuous Random Variables & The Normal Probability Distribution. Continuous Random Variable
Learning Objectives Continuous Random Variables & The Normal Probability Distribution 1. Understand characteristics about continuous random variables and probability distributions 2. Understand the uniform
More informationCHAPTER 3: FUNCTIONS IN 3-D
CHAPTER 3: FUNCTIONS IN 3-D 3.1 DEFINITION OF A FUNCTION OF TWO VARIABLES A function of two variables is a relation that assigns to every ordered pair of input values (x, y) a unique output value denoted
More informationCHAPTER 3: FUNCTIONS IN 3-D
CHAPTER 3: FUNCTIONS IN 3-D 3.1 DEFINITION OF A FUNCTION OF TWO VARIABLES A function of two variables is a relation that assigns to every ordered pair of input values (x, y) a unique output value denoted
More informationModule 3: Stand Up Conics
MATH55 Module 3: Stand Up Conics Main Math concepts: Conic Sections (i.e. Parabolas, Ellipses, Hyperbolas), nd degree equations Auxilliary ideas: Analytic vs. Co-ordinate-free Geometry, Parameters, Calculus.
More information