# Calculating Areas Section 6.1

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Calculating Areas Section 6.1 Dr. John Ehrke Department of Mathematics Fall 2012

2 Measuring Area By Slicing We first defined the integral in terms of the area bounded by a single function and the x-axis. Not surprisingly, integrals can also be used to measure areas of more general planar regions, bounded by two, three, and more graphs. Not surprisingly the basic idea comes back to approximations with rectangles. Example Approximate the area trapped between the curves y = sin x and y = cos x from x = 3π/4 and x = π/4. Remarks: The curves intersect at x = 3π/4 and x = π/4. The height of each rectangle is determined at its left edge, x = x i ; the height is cos(x i ) sin(x i ). The total area of ten rectangles is L 10, the left sum with ten equal subdivisions for the function cos x sin x on the interval [ 3π/4, π/4]. The area between the curves is the integral π/4 (cos x sin x) dx = π/4 Slide 2/31 Dr. John Ehrke Lecture 6 Fall 2012

3 The Area in Question Slide 3/31 Dr. John Ehrke Lecture 6 Fall 2012

4 What We are Really Calculating Slide 4/31 Dr. John Ehrke Lecture 6 Fall 2012

5 Finding Areas by Vertical Slices Example Find the area bounded by the graphs of y = 2x/3, y = x 2 2x 1 and the y-axis. Solution: We must start by finding the intersection between the graphs y = 2/3x and y = x 2 2x 1. This occurs at x = 3. We integrate the top and bottom curves and obtain 3 ( ) 2 Area = 0 3 x (x2 2x 1) dx = 6. Slide 5/31 Dr. John Ehrke Lecture 6 Fall 2012

6 A Slightly Harder Example Example Find the area of the region enclosed by x = y 2 and y = x 2. Solution: We start by finding where the curves intersect. This yields y 2 = y + 2 = y 2 y 2 = 0 = (y + 1)(y 2) = 0 from which we obtain y = 1, y = 2. Substituting into either equation we see that the corresponding x-values are x = 1 and x = 4, so the intersection points are (1, 1) and (4, 2). Vertical Slices: We split the region into two parts A 1, A 2 and find the area of each part separately. We have f (x) = x, g(x) = x, a = 0, b = 1 and so A 1 = x ( x) dx = 2 x dx = For A 2 we have f (x) = x, g(x) = x 2, a = 1, b = 4, so 4 19 A 2 = x (x 2) dx = 1 6. Horizontal Slices: If we integrate horizontally we do not have to split the region and r(y) = y + 2 and l(y) = y 2 so the area A for the region is given by 2 2 A = r(y) l(y) dx = y + 2 y 2 dy = Slide 6/31 Dr. John Ehrke Lecture 6 Fall 2012

7 Finding Areas by Horizontal Slices Example Find the area bounded by the graphs of y = x 2, x = 1 y 2, and the lines y = ±1. Solution: Because this region does not have consistent rectangles if we slice vertically, it makes sense to write the bounding functions in terms of y and integrate across the y-axis. This means 1 Area = (y + 2 (1 y 2 )) dy = 8/3. 1 Slide 7/31 Dr. John Ehrke Lecture 6 Fall 2012

8 Parametric Curves We have explored how to find areas by taking vertical slices and horizontal slices (rectangular thinking) and by taking circular wedges (polar thinking). In this lecture we will extend these techniques to deal with another variable change that leads to parametric equations. The basic idea behind parametric curves is that we write the x, y coordinates in terms of a parameter t (usually time). In symbols, x = x(t) y = y(t) The basic idea behind the formation of a parametric curve is that of a directed trajectory. Slide 8/31 Dr. John Ehrke Lecture 6 Fall 2012

9 Parameterizing a Line Segment Example Find a parametrization for the line segment with endpoints ( 2, 1) and (3, 5). Solution: Using ( 2, 1) as the starting point we create the parametric equations x = 2 + at, y = 1 + bt. We determine a and b so that the line will go through (3, 5) when t = 1. Solving each equation for t and equating the results we obtain This leads to Therefore, x + 2 a = y 1. b 3 = 2 + a = a = 5 5 = 1 + b = b = 4 x = 2 + 5t, y = 1 + 4t, 0 t 1 is a parametrization of the line segment with initial point ( 2, 1) and terminal point (3, 5). Slide 9/31 Dr. John Ehrke Lecture 6 Fall 2012

10 General Form of the Parametrization of a Line Parametrization of a Line Segment A line with starting point (α, β) and terminal point (γ, δ) can be parameterized by the equations x = γt + α(1 t), δt + β(1 t), 0 t 1. Solution: If we look at the previous example with starting point (α, β) = ( 2, 1) and terminal point (γ, δ) = (3, 5) we obtain the parametrization x = 3t 2(1 t), y = 5t + 1(1 t), 0 t 1. After collecting terms these equations agree with our previous answer. Slide 10/31 Dr. John Ehrke Lecture 6 Fall 2012

11 Standard Parameterizations Circle A parametrization of the circle x 2 + y 2 = a 2 is given by x = a cos t, y = a sin t, 0 t 2π. Ellipse A parametrization of the ellipse x2 a 2 + y2 = 1 is given by b2 x = a cos t, b sin t, 0 t 2π Standard Parametrization Every function y = f (x) admits a standard parametrization given by x = t, y = f (t) where the domain of t is determined by the domain of f (x). Similarly, the parametrization of the inverse of f (x) is given by x = f (t), y = t where the domain of t is determined by the range of f (x). Slide 11/31 Dr. John Ehrke Lecture 6 Fall 2012

12 Shapes Created Via Rolling Wheels Many fascinating curves are generated by points on rolling wheels. The path of light on the rim of a rolling wheel is a cycloid, which has parametric equations where a > 0. x = a(t sin t), y = a(1 cos t), t 0 Another equally interesting curve is the one generated by a circle with radius a/4 that rolls along the inside of a larger circle with radius a. The curve created in this case is called an astroid or hypocycloid, and its parametric equations are x = a cos 3 t, y = a sin 3 t, 0 t 2π. Example Find the rectangular forms of each of these parametric equations and sketch their graphs. Label the graphs with the parameter t. Slide 12/31 Dr. John Ehrke Lecture 6 Fall 2012

13 Graphs of the Cycloid and Asteroid Rectangular Equations Cycloid: x = a cos 1 ( 1 y a ) 2ay y 2 Asteroid: x 2/3 + y 2/3 = 1 Slide 13/31 Dr. John Ehrke Lecture 6 Fall 2012

14 Calculating Areas Enclosed by Parametric Curves As we are now no doubt aware of the area under a curve y = h(x) from a to b is given by the integral Area = b a h(x) dx. We think of h(x) as a height function, but if this function is defined by a parametric equation x = f (t) and y = g(t), then a simple substitution allows us to calculate area in the following way: b Area = y dx = g(t)f (t) dt where the limits of integration must be determined by inspection. a Slide 14/31 Dr. John Ehrke Lecture 6 Fall 2012

15 Area Enclosed by the Hypocycloid Example Find the area enclosed by the hypocycloid, x 2/3 + y 2/3 = 1. Recall the parametrization of this curve is given by x = a cos 3 t, y = a sin 3 t, 0 t 2π. Solution: Applying the formula from the previous slide we calculate the area enclosed in one quadrant and use symmetry to find the full answer. The area enclosed in quadrant I is given by Area = a 0 y dx = π/2 Multiplying the result by 4 for the full area gives 0 (a sin 3 t)(3a cos 2 t sin t) dt = 3 32 πa2. Area of a Hypocycloid = 3 8 πa2. Slide 15/31 Dr. John Ehrke Lecture 6 Fall 2012

16 A Slightly Harder Example Example Find the area bounded by the loop of the curve with parametric equation x = t 2, y = t 3 3t. Solution: The hardest part of finding areas involving parametric equations is determining the range over which t runs. We usually must resort to graphing to help as shown in the screen shots below. Slide 16/31 Dr. John Ehrke Lecture 6 Fall 2012

17 Solution From the equation it is clear that when t = 0, (x, y) = (0, 0). This implies the graph begins at the origin, but to form the loop we need to know the values of t that result in the intersection. This would clearly be a value of t where y = 0 based on the graph, so we solve t 3 3t = 0 = t(t 2 3) = 0 = t = ± 3. We can recover the area enclosed by finding the top (or bottom half area) and using symmetry. The area enclosed by the curve is given by Area = y dx = (t 3 3t)(2t) dt = Slide 17/31 Dr. John Ehrke Lecture 6 Fall 2012

18 National Curve Bank The National Curve Bank is a website supported by the National Science Foundation (NSF) and the Beckman foundation and is run by California State University (LA). The website is located at and is home to a wide array of animations and graphs of curves arising in all sorts of mathematical applications. Slide 18/31 Dr. John Ehrke Lecture 6 Fall 2012

19 Polar Coordinates One of the most important coordinate systems we work with in mathematics is the polar coordinate system. Polar coordinates are inspired by our work with trigonometric functions and work best when trying to analyze annular regions (or regions involving circles or ellipses). The basic idea is that given our familiarity with sin θ and cos θ we are able to completely redefine the way we think of plotting points in the plane. Our regular coordinate system is called a rectangular coordinate system. In this lecture, we will introduce the polar coordinate system and explore its application to finding areas of annular regions. r = distance to the origin x = r cos θ r = ± x 2 y 2 θ = standard angle rotation y = r sin θ θ = tan 1 ( y ) x Slide 19/31 Dr. John Ehrke Lecture 6 Fall 2012

20 Plotting Points in Polar Coordinates Consider plotting the point (x, y) = (1, 1) in polar coordinates. What are the coordinates which describe this exactly? The problem with the term exactly implies there is only one answer, and when it comes to polar coordinates that is simply just not the case. In this small example, the point (1, 1) in rectangular coordinates can be described by just to name a few. (r, θ) = ( 2, 3π/4), ( 2, 7π/4), ( 2, π/4) So how do we handle this ambiguity? The typical convention for polar coordinates is that the variable r and θ obey the following: 1 0 r < 2 π θ π 3 0 θ 2π As you can see there is not even a consensus on the range for θ, but we ll generally use 0 θ 2π unless otherwise indicated. Slide 20/31 Dr. John Ehrke Lecture 6 Fall 2012

21 Coordinate Comparisons Figure: Rectangular Coordinate System Figure: Polar Coordinate System Slide 21/31 Dr. John Ehrke Lecture 6 Fall 2012

22 From Polar to Rectangular and Back Again Consider the function in the polar coordinate system determined by the graph below. What is the equation in polar coordinates? In rectangular coordinates? Slide 22/31 Dr. John Ehrke Lecture 6 Fall 2012

23 Solution continued... Looking at the graph on the previous slide we should notice this as a circle of radius a centered at (a, 0). In rectangular coordinates, this gives (x a) 2 + y 2 = a 2. We now look to convert this expression to polar coordinates. Notice that if we expand the left hand side we obtain This means x 2 2ax + a 2 + y 2 = a 2. x 2 + y 2 2ax = 0 = r 2 2a(r cos θ) = 0. Finally, we obtain the expression in polar coordinates, r = r(θ), given by r(θ) = 2a cos θ. You should note that when θ = 0 we are at the point (2a, 0) and when θ = ±π/2 we are at the point (0, 0). This implies the angle θ takes on the range θ [ π/2, π/2] if we want the angle drawn out to use positive radii the whole way through which was one of our conventions stated earlier. Slide 23/31 Dr. John Ehrke Lecture 6 Fall 2012

24 Calculating Area Using Polar Coordinates In order to get a feel for how to calculate areas using polar coordinates we consider the basic problem of finding the area of a circle of radius a. The basic approach involves summing wedges as opposed to rectangles like we did with Riemann sums. It is pretty easy to find the area of this pie wedge as it is simply a fraction of the total area of the circle which we know to be A = πa 2. Looking at A, we have A = θ 2π πa2. This means that the formula for the area of these wedges is given by A = 1 2 a2 θ. This is also the formula for the area of a sector of a circle. Slide 24/31 Dr. John Ehrke Lecture 6 Fall 2012

25 The Variable Pie As we know, not all regions in the plane are nice circles like this, but we ve established a basic manner of thought here, that can actually pay dividends when applied to a more general situation. Consider the quadrant of a pie with variable boundary given by r = r(θ). How do we find the area of this region? From our previous experience with the circle, we know that in terms of r(θ), the area A of the adjusted wedge is A = 1 2 r2 θ. Summing over all such wedges and letting θ 0, we obtain the integral formula for the total area over the annular region θ 1 θ θ 2, θ2 1 θ 1 2 r(θ)2 dθ. Slide 25/31 Dr. John Ehrke Lecture 6 Fall 2012

26 An Example Example Find the area of the circle centered at (a, 0) of radius a given by r(θ) = 2a cos θ. Solution: It is important to note that this is the same figure considered earlier in this lecture. In this case, the variable radius is given by r(θ) = 2a cos θ, so applying our polar area formula, we have π/2 1 Area = π/2 2 (2a cos θ)2 dθ π/2 = 2a 2 cos 2 θ dθ π/2 π/2 = 2a cos(2θ) dθ π/2 2 ( = a 2 θ + 1 ) π/2 2 sin(2θ) = πa 2 Which is exactly the area we expected from basic geometry. π/2 Slide 26/31 Dr. John Ehrke Lecture 6 Fall 2012

27 An Example With Symmetries Example Find the entire area enclosed by the curve r = 2 sin 3θ. Solution: We will begin by drawing the curve (using the polar graph on the next slide and plotting points). In fact, in all area problems in polar coordinates a sketch should be made. The curve is called a rose curve. A single petal is described completely as θ goes from 0 π/3. This means we can use symmetries and find the area of this loop only. We have Integration yields, A 3 = 1 2 π/3 π/3 (2 sin 3θ) 2 dθ = A = [ θ A = ] π/3 sin 6θ = π cos 6θ 2 dθ. Slide 27/31 Dr. John Ehrke Lecture 6 Fall 2012

28 Slide 28/31 Dr. John Ehrke Lecture 6 Fall 2012

29 Intersecting Polar Curves Example Find the area inside the circle r = 5 cos θ and outside the curve r = 2 + cos θ. Solution: The two curves intersect when 5 cos θ = 2 + cos θ cos θ = 1/2. This means θ = ±π/3. The area inside 5 cos θ and outside 2 + cos θ is given by A = 1 2 π/3 π/3(5 cos θ) 2 dθ 1 2 π/3 π/3 (2 + cos θ) 2 dθ. We observe that both curves are symmetric to the x-axis and the integrals may be combined since the limits of integration are the same. This gives, A = π/3 0 [25 cos 2 θ (4 + 4 cos θ + cos 2 θ)] dθ = [8θ + 6 sin 2θ 4 sin θ] π/3 0 = 8π Slide 29/31 Dr. John Ehrke Lecture 6 Fall 2012

30 A Harder Example Example Find the area that lies inside the larger loop and outside the smaller loop of the limacon, r = 1 + cos θ. 2 Slide 30/31 Dr. John Ehrke Lecture 6 Fall 2012

31 Solution... The trick to finding the solution in this case is figuring out the value of θ for which certain parts are drawn and what the radius looks like for those parts. We use the symmetry across the x-axis by finding only the area trapped in the top half and then multiplying by 2. For 0 θ 2π/3, the top lobe is drawn out. We need to then subtract the area contained in the smaller loop which is drawn out for π θ 4π/3. All together (remembering to multiply by 2) we have [ 2π/3 ] 1 4π/3 1 A = r2 dθ π 2 r2 dθ 2π/3 ( ) 2 1 4π/3 = 2 + cos θ dθ π ( ) cos θ dθ Slide 31/31 Dr. John Ehrke Lecture 6 Fall 2012

### Section 10.7 Parametric Equations

299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x- (rcos(θ), rsin(θ)) and y-coordinates on a circle of radius r as a function of

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### Example 1. Example 1 Plot the points whose polar coordinates are given by

Polar Co-ordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points

### Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

### This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,

### Intersections of Polar Curves

Intersections of Polar Curves The purpose of this supplement is to find a method for determining where graphs of polar equations intersect each other. Let s start with a fairly straightforward example.

### PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

### 10 Polar Coordinates, Parametric Equations

Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry 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

### Double integrals. Notice: this material must not be used as a substitute for attending the lectures

ouble integrals Notice: this material must not be used as a substitute for attending the lectures . What is a double integral? Recall that a single integral is something of the form b a f(x) A double integral

### 7.3 Volumes Calculus

7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are

### Math 1B, lecture 5: area and volume

Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Area in Polar Coordinates

Area in Polar Coordinates If we have a circle of radius r, and select a sector of angle θ, then the area of that sector can be shown to be 1. r θ Area = (1/)r θ As a check, we see that if θ =, then the

### Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

### National Quali cations 2015

H National Quali cations 05 X77/76/ WEDNESDAY, 0 MAY 9:00 AM 0:0 AM Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only

### POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

### SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions

SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions All questions in the Math Level 1 and Math Level Tests are multiple-choice questions in which you are asked to choose the

### Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

### Graphs of Polar Equations

Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate

### GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

### Integration Involving Trigonometric Functions and Trigonometric Substitution

Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving

### MULTIPLE INTEGRATION NOTES AND EXERCISES. Antony Foster

MULTIPL INTGRATION NOTS AN XRCISS by Antony Foster Overview In this discussion we consider the integral of a function of two variables z f(x, y) over a region in the xy-plane (-space) and the integral

### GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

### Area and Arc Length in Polar Coordinates

Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly

### Solutions to Vector Calculus Practice Problems

olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x

### 1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

### Mark Howell Gonzaga High School, Washington, D.C.

Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION

MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 ASIC INTEGRATION This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

### Week #15 - Word Problems & Differential Equations Section 8.1

Week #15 - Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### Homework from Section Find two positive numbers whose product is 100 and whose sum is a minimum.

Homework from Section 4.5 4.5.3. Find two positive numbers whose product is 100 and whose sum is a minimum. We want x and y so that xy = 100 and S = x + y is minimized. Since xy = 100, x = 0. Thus we have

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Solutions to Practice Problems for Test 4

olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,

### Techniques of Integration

CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration

### (b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation

(1) Consider the function y = f(x) =e x on the interval [, 1]. (a) Find the area under the graph of this function over this interval using the Fundamental Theorem of Calculus. (b) Subdivide the interval

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### 3. Double Integrals 3A. Double Integrals in Rectangular Coordinates

3. Double Integrals 3A. Double Integrals in ectangular Coordinates 3A-1 Evaluate each of the following iterated integrals: c) 2 1 1 1 x 2 (6x 2 +2y)dydx b) x 2x 2 ydydx d) π/2 π 1 u (usint+tcosu)dtdu u2

### Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

### 4 More Applications of Definite Integrals: Volumes, arclength and other matters

4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Graphing Linear Equations in Two Variables

Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the

### Trigonometry Lesson Objectives

Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

### MATH 132: CALCULUS II SYLLABUS

MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

### Plotting Polar Curves We continue to study the plotting of polar curves. Recall the family of cardioids shown last time.

Plotting Polar Curves We continue to study the plotting of polar curves. Recall the family of cardioids shown last time. r = 1 cos(θ) r = 1 + cos(θ) r = 1 + sin(θ) r = 1 sin(θ) Now let us look at a similar

### Parametric Curves, Vectors and Calculus. Jeff Morgan Department of Mathematics University of Houston

Parametric Curves, Vectors and Calculus Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu Online Masters of Arts in Mathematics at the University of Houston http://www.math.uh.edu/matweb/grad_mam.htm

2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

### Additional Topics in Math

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Triple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains.

Triple integrals in Cartesian coordinates (Sect. 5.4 Review: Triple integrals in arbitrar domains. s: Changing the order of integration. The average value of a function in a region in space. Triple integrals

### Double 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

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

### angle Definition and illustration (if applicable): a figure formed by two rays called sides having a common endpoint called the vertex

angle a figure formed by two rays called sides having a common endpoint called the vertex area the number of square units needed to cover a surface array a set of objects or numbers arranged in rows and

### AP CALCULUS AB 2008 SCORING GUIDELINES

AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

### TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### 55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

### Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)

Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.

### correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

### Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

### Draft copy. Circles, cylinders and prisms. Circles

12 Circles, cylinders and prisms You are familiar with formulae for area and volume of some plane shapes and solids. In this chapter you will build on what you learnt in Mathematics for Common Entrance

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### 4.1: Angles and Radian Measure

4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is

### x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3

CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract -

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### Pre-Calculus Review Problems Solutions

MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry Pre-Calculus Review Problems Solutions Problem 1. Give equations for the following lines in both point-slope and slope-intercept form. (a) The

### 5.2 Unit Circle: Sine and Cosine Functions

Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and

### Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### Section 10-5 Parametric Equations

88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

### Angles and Their Measure

Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11 - NQF LEVEL 3 OUTCOME 3 - ROTATING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11 - NQF LEVEL 3 OUTCOME 3 - ROTATING SYSTEMS TUTORIAL 1 - ANGULAR MOTION CONTENT Be able to determine the characteristics

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### 4.4 CURVE SKETCHING. there is no horizontal asymptote. Since sx 1 l 0 as x l 1 and f x is always positive, we have

SECTION 4.4 CURE SKETCHING 4.4 CURE SKETCHING EXAMPLE A Sketch the graph of f x. sx A. Domain x x 0 x x, B. The x- and y-intercepts are both 0. C. Symmetry: None D. Since x l sx there is no horizontal

### Practice Problems for Exam 1 Math 140A, Summer 2014, July 2

Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other

### Readings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.1-2.2. Professor Christopher Hoffman Math 124

Readings this week 1 Parametric Equations Supplement 2 Section 10.1 3 Sections 2.1-2.2 Precalculus Review Quiz session Thursday equations of lines and circles worksheet available at http://www.math.washington.edu/

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### Lab M1: The Simple Pendulum

Lab M1: The Simple Pendulum Introduction. The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are usually regarded as the beginning of

### The Stern-Gerlach Experiment

Chapter The Stern-Gerlach Experiment Let us now talk about a particular property of an atom, called its magnetic dipole moment. It is simplest to first recall what an electric dipole moment is. Consider

### Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

### Volumes of Revolution

Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students

### Arrangements And Duality

Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other