Name Class. (a) (b) (c) 2. Find the volume of the solid formed by revolving the region bounded by the graphs of
|
|
|
- Lauren McCoy
- 5 years ago
- Views:
Transcription
1 Applications of Integration Test Form A. Determine the area of the region bounded by the graphs of y x 4x and y x 4. (a) 9 9 (b) 6 (c). Find the volume of the solid formed by revolving the region bounded by the graphs of y x, y and x about the x-axis. 7 4 (a) 7 (b) 7 (c) 7. Which of the following integrals represents the volume of the solid formed by revolving the region bounded by the graphs of y x, y and x about the line x? (a) yy (b) dy x dx (c) xx dx 4. Find the volume of the solid formed by revolving the region bounded by the graphs of y x 4x and y about the y-axis. y dy 544 (a) (b) (c) Identify the definite integral that represents the arc length of the curve y x over the interval,. (a) (b) (c) x dx x dx 4x dx x dx
2 Test Bank 6. Identify the definite integral that represents the area of the surface formed by revolving the graph of fx x on the interval, about the x axis. (a) x x 4 dx (b) x 4x dx (c) y y dy x 4x dx 7. A force of pounds stretches a spring 4 foot on an exercise machine. Find the work done in stretching the spring foot. (a) ft-lb (b) 5 ft-lb (c) ft-lb 4 ft-lb 9. Find the x-coordinate of the centroid of the region bounded by the graphs of y x, y and x. (a) 7 (b) 4 (c) 4 7
3 Applications of Integration Test Form B. Determine the area of the region bounded by the graphs of y x x and y. (a) 4 9 (b) (c) 4. Find the volume of the solid formed by revolving the region bounded by the graphs of y x, x and y about the y-axis (a) 5 (b) 7 (c) Which of the following integrals represents the volume of the solid formed by revolving the region bounded by the graphs of y x, y and x about the line y? (a) y y dy (b) x dx (c) x dx y y dy 4. Find the volume of the solid formed by revolving the region bounded by the graphs of and y about the y-axis. y x 64 (a) (b) (c) 5 5. Identify the definite integral that represents the arc length of the curve y x over the interval,. (a) ln x dx (b) x dx (c) x 4 dx x x 4 dx
4 Test Bank 6. Identify the definite integral that represents the area of the surface formed by revolving the graph of fx x on the interval, about the y-axis. (a) x 9x 4 dx (b) x 9x 4 dx (c) x x dx x x dx 7. A force of pounds compresses a spring inches. How much work is done on compressing the spring 6 inches? (a) 4 in-lb (b) 6 in-lb (c) 4 in-lb in-lb. Find the y-coordinate of the centroid of the region bounded by the graphs of y x, y and x. 4 (a) 5 (b) 5 (c) 4 5
5 4 Applications of Integration Test Form C A graphing calculator is needed for some problems. b. The integral sin x e x dx computes the area of a region between two curves. a Use a graphing calculator to estimate the value of a. (a) (b). (c)..6. Use a graphing calculator to graph the region bounded by the graphs of y x x, y, and x. Then use calculus to compute the volume of the solid formed by revolving this region about the x-axis. 4 7 (a) (b) (c) Use the integration capabilities of a graphing calculator to approximate the volume of the solid formed by revolving the region bounded by the graphs of y sin x and y in the interval, about the y-axis. Round your answer to three decimal places. (a).6 (b) 4.95 (c) Use the integration capabilities of a graphing calculator to approximate the arc length of the graph of fx cos xon the interval,. Round your answer to three decimal places. (a) (b). (c) Use the integration capabilities of a graphing calculator to approximate the area of the surface formed by revolving the graph of fx sin x on the interval, about the y-axis. Round your answer to three decimal places. (a) 6. (b) 7.74 (c) A force of 5 pounds compresses a spring 5 inches from its natural length. Find the work done in compressing the spring additional inches. [Units are in inch-pounds.] (a),5 (b) 5 (c),
6 Test Bank 5 7. Use the Theorem of Pappus to find the volume of the solid of revolution of the torus formed by revolving the circle x y 4 about the x-axis. (a) (b) (c) Find the centroid of the region bounded by the graphs of fx x and gx 4 (a) 4 (b), (c) 6, 6, 45, x. 4
7 6 Applications of Integration Test Form D. Find the area of the region bounded by the graphs of y x and x y 5.. Find the volume of the solid formed by revolving the region bounded by the graphs of y e x, y, x, and x about the x-axis.. Use the shell method to set up the integral that represents the volume of the solid formed by revolving the region bounded by the graphs of y x and x y 5 about the line y. (Do not evaluate the integral.) 4. Find the volume of the solid formed by revolving the region bounded by the graphs of y x and y 4 about the x-axis. 5. Write the definite integral that represents the arc length of one period of the curve (Do not evaluate the integral.) y sin x. 6. Write the definite integral that represents the area of the surface formed by revolving the graph of fx x on the interval, 4 about the y-axis. (Do not evaluate the integral.) 7. A force of 4 pounds compresses a -inch spring by inches. How much work is done in compressing the spring to a final length of 4 inches?. Find the centroid of the region bounded by the graphs of y x, y, and x.
8 Test Bank 7 Test Form E A graphing calculator is needed for some problems. b a. The integral ln x x dx computes the area of a region between two curves. a. Use a graphing calculator to graph the curves y ln x and y x. b. Use the graph to estimate the values of a and b. c. Use the integration capabilities of a graphing calculator to approximate the area of the region. Round your answer to three decimal places.. Use the integration capabilities of a graphing calculator to approximate the volume of the solid formed by revolving the region bounded by the graphs of y ln x, x y, and the x-axis about the x-axis. Round your answer to three decimal places.. Consider the region bounded by the graphs of y y x, and x. x, a. Sketch the region described. b. Find the point(s) of intersection. c. Calculate the volume of the solid formed when this region is revolved about the line x. 4. A telephone wire suspended between two poles forms a catenary modeled by the equation x y cosh, 5 x 5 where x and y are measured in feet. Approximate the length of the suspended cable if the poles are feet apart. 5. Neglecting air resistance, determine the work done in propelling a -ton satellite to a height of miles above the surface of the earth. 6. A circular observation window on a cruise ship has a radius of foot and the center of the window is feet below water level. What is the fluid force on the window? (Use 64 lbft as the weight-density of sea water.) 7. Consider the plane region bounded by the graphs of fx x and gx. a. Calculate the moment about the x-axis. b. Find the centroid of the region. c. Find the volume of the solid formed by revolving the region about the x-axis using the Theorem of Pappus. d. Use disc method to find the volume described in part c.. Use the integration capabilities of a graphing calculator to approximate the centroid of the region bounded by the graphs y ln x and y. Round your answer to three decimal places.
LECTURE 3-1 AREA OF A REGION BOUNDED BY CURVES
7 CALCULUS II DR. YOU 98 LECTURE 3- AREA OF A REGION BOUNDED BY CURVES If y = f(x) and y = g(x) are continuous on an interval [a, b] and f(x) g(x) for all x in [a, b], then the area of the region between
5/27/12. 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.
Topic 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
For Test #1 study these problems, the examples in your notes, and the homework.
Mth 74 - Review Problems for Test Test covers Sections 6.-6.5, 7. and 7. For Test # study these problems, the examples in your notes, and the homework.. The base of a solid is the region inside the circle
CHAPTER 6: APPLICATIONS OF INTEGRALS
(Exercises for Section 6.1: Area) E.6.1 CHAPTER 6: APPLICATIONS OF INTEGRALS SECTION 6.1: AREA 1) For parts a) and b) below, in the usual xy-plane i) Sketch the region R bounded by the graphs of the given
AP * 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,
Exam 3 SCORE. MA 114 Exam 3 Spring Section and/or TA:
MA 114 Exam 3 Spring 217 Exam 3 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test.
6.2 Volumes by Disks, Washers, and Cross-Sections
6.2 Volumes by Disks, Washers, and Cross-Sections General Principle: Disks Take slices PERPENDICULAR to axis of rotation and rotate around that axis. About x-axis: About y-axis: 1 Examples: Set up integrals
Math 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
Chapter 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
Area 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:
Name: 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
The 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
During the timed portion for Part A, you may work only on the problems in Part A.
SECTION II Time: hour and 30 minutes Percent of total grade: 50 Part A: 45 minutes, 3 problems (A graphing calculator is required for some problems or parts of problems.) During the timed portion for Part
AP 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
Conics, Parametric Equations, and Polar Coordinates. Copyright Cengage Learning. All rights reserved.
10 Conics, Parametric Equations, and Polar Coordinates Copyright Cengage Learning. All rights reserved. 10.5 Area and Arc Length in Polar Coordinates Copyright Cengage Learning. All rights reserved. Objectives
Conics, Parametric Equations, and Polar Coordinates. Copyright Cengage Learning. All rights reserved.
10 Conics, Parametric Equations, and Polar Coordinates Copyright Cengage Learning. All rights reserved. 10.5 Area and Arc Length in Polar Coordinates Copyright Cengage Learning. All rights reserved. Objectives
MATH 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
Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes
1 of 11 1) Give f(g(1)), given that Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes 2) Find the slope of the tangent line to the graph of f at x = 4, given that 3) Determine
x + 2 = 0 or Our limits of integration will apparently be a = 2 and b = 4.
QUIZ ON CHAPTER 6 - SOLUTIONS APPLICATIONS OF INTEGRALS; MATH 15 SPRING 17 KUNIYUKI 15 POINTS TOTAL, BUT 1 POINTS = 1% Note: The functions here are continuous on the intervals of interest. This guarantees
AB Student Notes: Area and Volume
AB Student Notes: Area and Volume An area and volume problem has appeared on every one of the free response sections of the AP Calculus exam AB since year 1. They are straightforward and only occasionally
MA 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
Volume by Slicing (Disks & Washers)
Volume by Slicing Disks & Washers) SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 6. of the recommended textbook or the equivalent chapter in
Plane Curve [Parametric Equation]
![Plane Curve [Parametric Equation]](/thumbs/89/98510023.jpg "Plane Curve [Parametric Equation]")
Plane Curve [Parametric Equation] Bander Almutairi King Saud University December 1, 2015 Bander Almutairi (King Saud University) Plane Curve [Parametric Equation] December 1, 2015 1 / 8 1 Parametric Equation
V = 2πx(1 x) dx. x 2 dx. 3 x3 0
Wednesday, September 3, 215 Page 462 Problem 1 Problem. Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the region (y = x, y =, x = 2)
Unit 4. Applications of integration
Unit 4. Applications of integration 4A. Areas between curves. 4A-1 Find the area between the following curves a) y = 2x 2 and y = 3x 1 b) y = x 3 and y = ax; assume a > 0 c) y = x + 1/x and y = 5/2. d)
minutes/question 26 minutes
st Set Section I (Multiple Choice) Part A (No Graphing Calculator) 3 problems @.96 minutes/question 6 minutes. What is 3 3 cos cos lim? h hh (D) - The limit does not exist.. At which of the five points
Name Find the area the shaded region.
Berkele Cit College Due: HW - Chapter 6 - Applications of Integration Name Find the area of the shaded region. 1) = - + + 16 0 = 4 0 1 10 1) - -4 - - -1-1 4-10 -1-0 - -0 ) 6 = - 1 ) 4 = - 4 + 1 - -1 1
The 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
MATH 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
NAME: 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)
MATH 1112 Trigonometry Final Exam Review
MATH 1112 Trigonometry Final Exam Review 1. Convert 105 to exact radian measure. 2. Convert 2 to radian measure to the nearest hundredth of a radian. 3. Find the length of the arc that subtends an central
Name Date Period. Worksheet 6.3 Volumes Show all work. No calculator unless stated. Multiple Choice
Name Date Period Worksheet 6. Volumes Show all work. No calculator unless stated. Multiple Choice. (Calculator Permitted) The base of a solid S is the region enclosed by the graph of y ln x, the line x
Unit 4. Applications of integration
18.01 EXERCISES Unit 4. Applications of integration 4A. Areas between curves. 4A-1 Find the area between the following curves a) y = 2x 2 and y = 3x 1 b) y = x 3 and y = ax; assume a > 0 c) y = x + 1/x
Integration. Edexcel GCE. Core Mathematics C4
Edexcel GCE Core Mathematics C Integration Materials required for examination Mathematical Formulae (Green) Items included with question papers Nil Advice to Candidates You must ensure that your answers
Chapter 7 curve. 3. x=y-y 2, x=0, about the y axis. 6. y=x, y= x,about y=1
Chapter 7 curve Find the volume of the solid obtained by rotating the region bounded by the given cures about the specified line. Sketch the region, the solid, and a typical disk or washer.. y-/, =, =;
Review for Applications of Definite Integrals Sections
Review for Applications of Definite Integrals Sections 6.1 6.4 Math 166 Iowa State University http://orion.math.iastate.edu/dstolee/teaching/15-166/ September 4, 2015 1. What type of problem: Volume? Arc
Volume Worksheets (Chapter 6)
Volume Worksheets (Chapter 6) Name page contents: date AP Free Response Area Between Curves 3-5 Volume b Cross-section with Riemann Sums 6 Volume b Cross-section Homework 7-8 AP Free Response Volume b
9.1 Centroids by Integration
9.1 Centroids b Integration 9.1 Centroids b Integration Procedures and Strategies, page 1 of 2 Procedures and Strategies for Solving Problems Involving Calculating Centroids b Integration = f () (, ) 1.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. 1)
Math 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
AP 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
y = 4x + 2, 0 x 1 Name: Class: Date: 1 Find the area of the region that lies under the given curve:
Name: Class: Date: 1 Find the area of the region that lies under the given curve: y = 4x + 2, 0 x 1 Select the correct answer. The choices are rounded to the nearest thousandth. 8 Find the volume of the
Section 7.2 Volume: The Disk Method
Section 7. Volume: The Disk Method White Board Challenge Find the volume of the following cylinder: No Calculator 6 ft 1 ft V 3 1 108 339.9 ft 3 White Board Challenge Calculate the volume V of the solid
Multiple Integrals. max x i 0
Multiple Integrals 1 Double Integrals Definite integrals appear when one solves Area problem. Find the area A of the region bounded above by the curve y = f(x), below by the x-axis, and on the sides by
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:
Applications of Integration. Copyright Cengage Learning. All rights reserved.
Applications of Integration Copyright Cengage Learning. All rights reserved. 1 Arc Length and Surfaces of Revolution Copyright Cengage Learning. All rights reserved. 2 Objectives Find the arc length of
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Review for Test 2 MATH 116 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the right triangle. If two sides are given, give angles in degrees and
I 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
Math 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.
Chapter 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
Math 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
Contents. 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
Find the volume of a solid with regular cross sections whose base is the region between two functions
Area Volume Big Ideas Find the intersection point(s) of the graphs of two functions Find the area between the graph of a function and the x-axis Find the area between the graphs of two functions Find the
6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.
Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct
x=2 26. y 3x Use calculus to find the area of the triangle with the given vertices. y sin x cos 2x dx 31. y sx 2 x dx
4 CHAPTER 6 APPLICATIONS OF INTEGRATION 6. EXERCISES 4 Find the area of the shaded region.. =5-. (4, 4) =. 4. = - = (_, ) = -4 =œ + = + =.,. sin,. cos, sin,, 4. cos, cos, 5., 6., 7.,, 4, 8., 8, 4 4, =_
5/27/12. Objectives. Plane Curves and Parametric Equations. Sketch the graph of a curve given by a set of parametric equations.
Objectives Sketch the graph of a curve given by a set of parametric equations. Eliminate the parameter in a set of parametric equations. Find a set of parametric equations to represent a curve. Understand
MAT 1475 Final Exam Review Problems
MAT1475 Final Review Spring 2016 Spring 2016 MAT 1475 Final Exam Review Problems Revised by Prof. Kostadinov, Fall 2015, Fall 2014, Fall 2013, Fall 2012, Fall 2011, Fall 2010 Revised by Prof. Africk and
12 m. 30 m. The Volume of a sphere is 36 cubic units. Find the length of the radius.
NAME DATE PER. REVIEW #18: SPHERES, COMPOSITE FIGURES, & CHANGING DIMENSIONS PART 1: SURFACE AREA & VOLUME OF SPHERES Find the measure(s) indicated. Answers to even numbered problems should be rounded
AREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surface of revolution is formed when a curve is rotated about a line. Such a surface is the lateral boundar of a solid of revolution of the tpe discussed
Direction 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
AP 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
5 Applications of Definite Integrals
5 Applications of Definite Integrals The previous chapter introduced the concepts of a definite integral as an area and as a limit of Riemann sums, demonstrated some of the properties of integrals, introduced
AP 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.
Differentiation and Integration
Edexcel GCE Core Mathematics C Advanced Subsidiary Differentiation and Integration Materials required for examination Mathematical Formulae (Pink or Green) Items included with question papers Nil Advice
Unit 2: Functions and Graphs
AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to exactly one element in the range. The domain is the set of all possible
Questions Q1. (a) Find the values of the constants A, B and C. (4) b) Hence find
Questions Q1. (a) Find the values of the constants A, B and C. (4) b) Hence find (ii) Find, leaving your answer in the form a + ln b, where a and b are constants. (6) (Total 10 marks) Q2. (a) Find the
Practice problems from old exams for math 233 William H. Meeks III December 21, 2009
Practice problems from old exams for math 233 William H. Meeks III December 21, 2009 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These
Cables have been used in the design
L A B 14 SUSPENSION BRIDGES Parabolas Cables have been used in the design of many different types of structures. They have been used in the design of suspension bridges such as New York s Verrazano Narrows
4.1 Radian and Degree Measure
4.1 Radian and Degree Measure Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 1 / 27 4.1 Radian and Degree Measure 1 Angles Accelerated Pre-Calculus
Graphs of Equations. MATH 160, Precalculus. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Graphs of Equations
Graphs of Equations MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: sketch the graphs of equations, find the x- and y-intercepts
Math 251 Quiz 5 Fall b. Calculate. 2. Sketch the region. Write as one double integral by interchanging the order of integration: 2
Math 251 Quiz 5 Fall 2002 1. a. Calculate 5 1 0 1 x dx dy b. Calculate 1 5 1 0 xdxdy 2. Sketch the region. Write as one double integral by interchanging the order of integration: 0 2 dx 2 x dy f(x,y) +
Section 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.
1 Section 7.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint
f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim
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.3
2.2 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
Applications of Integration
Week 12. Applications of Integration 12.1.Areas Between Curves Example 12.1. Determine the area of the region enclosed by y = x 2 and y = x. Solution. First you need to find the points where the two functions
Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 28 / 46
Polar Coordinates Polar Coordinates: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ)
P1 REVISION EXERCISE: 1
P1 REVISION EXERCISE: 1 1. Solve the simultaneous equations: x + y = x +y = 11. For what values of p does the equation px +4x +(p 3) = 0 have equal roots? 3. Solve the equation 3 x 1 =7. Give your answer
Multivariate Calculus: Review Problems for Examination Two
Multivariate Calculus: Review Problems for Examination Two Note: Exam Two is on Tuesday, August 16. The coverage is multivariate differential calculus and double integration. You should review the double
Volumes 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
MAT01B1: Surface Area of Solids of Revolution
MAT01B1: Surface Area of Solids of Revolution Dr Craig 02 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
Calculus 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
Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 27 / 45
: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations
HSC Mathematics - Extension 1. Workshop E2
HSC Mathematics - Extension Workshop E Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong Moss
= 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
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Calculus III-Final review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the corresponding position vector. 1) Define the points P = (-,
Volume by Slicing (Disks & Washers)
Volume by Slicing (Disks & Washers) SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 6.2 of the recommended textbook (or the equivalent chapter
m x x Assignment #2 MAT121 Summer 2017 NAME:
Assignment # MAT11 Summer 017 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also
Applications of Integration. Copyright Cengage Learning. All rights reserved.
Applications of Integration Copyright Cengage Learning. All rights reserved. Volume: The Disk Method Copyright Cengage Learning. All rights reserved. Objectives Find the volume of a solid of revolution
PRECALCULUS MATH Trigonometry 9-12
1. Find angle measurements in degrees and radians based on the unit circle. 1. Students understand the notion of angle and how to measure it, both in degrees and radians. They can convert between degrees
Tuesday 22 January 2008 Afternoon Time: 1 hour 30 minutes
Paper Reference(s) 6666/0 Edexcel GCE Core Mathematics C4 Advanced Level Tuesday 22 January 2008 Afternoon Time: hour 30 minutes Materials required for examination Mathematical Formulae (Green) Items included
Multivariate Calculus Review Problems for Examination Two
Multivariate Calculus Review Problems for Examination Two Note: Exam Two is on Thursday, February 28, class time. The coverage is multivariate differential calculus and double integration: sections 13.3,
Name Modeling with Geometry 3.1H
Name Modeling with Geometry 3.1H Ready, Set, Go! Ready Topic: Finding the trigonometric ratios in a triangle Use the given measures on the triangles to write the indicated trigonometry value. Leave answers
MA 113 Calculus I Fall 2015 Exam 2 Tuesday, 20 October Multiple Choice Answers. Question
MA 113 Calculus I Fall 2015 Exam 2 Tuesday, 20 October 2015 Name: Section: Last digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions (ten
sin30 = sin60 = cos30 = cos60 = tan30 = tan60 =
Precalculus Notes Trig-Day 1 x Right Triangle 5 How do we find the hypotenuse? 1 sinθ = cosθ = tanθ = Reciprocals: Hint: Every function pair has a co in it. sinθ = cscθ = sinθ = cscθ = cosθ = secθ = cosθ
Mid-Chapter Quiz: Lessons 9-1 through 9-3
Graph each point on a polar grid. 1. A( 2, 45 ) 3. Because = 45, locate the terminal side of a 45 angle with the polar axis as its initial side. Because r = 2, plot a point 2 units from the pole in the
Name: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.
. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f (x, y) = x y, x + y = 8. Set up the triple integral of an arbitrary continuous function
EXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR
EXERCISE SET 10. STUDENT MATD 090 DUE DATE: INSTRUCTOR You have studied the method known as "completing the square" to solve quadratic equations. Another use for this method is in transforming the equation
MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals)
MA 43 Calculus III Fall 8 Dr. E. Jacobs Assignments Reading assignments are found in James Stewart s Calculus (Early Transcendentals) Assignment. Spheres and Other Surfaces Read. -. and.6 Section./Problems
Algebra 2B CH 5. WYNTK & TEST Algebra 2B What You Need to Know , Test
Algebra 2B CH 5 NAME: WYNTK 5.1 5.3 & 5.7 5.8 TEST DATE: HOUR: Algebra 2B What You Need to Know 5.1 5.3, 5.7-5.8 Test A2.5.1.2 Be able to use transformations to graph quadratic functions and answer questions.