The document provides an introduction to integral calculus and integration. It discusses:
- The concept of integration as reversing the process of differentiation and as a summation function.
- Standard forms of integrals including kxn with n ≠ -1, k/x, and kanx with a > 0.
- Applying integration to determine the area under a curve, between ordinates, or a region bounded by curves and axes.
- Worked examples of evaluating indefinite integrals using standard forms and the area under curves using the mid-ordinate rule with trapezoids.
Integral Calculus Anti Derivatives reviewerJoshuaAgcopra
This document provides an overview of integration concepts and formulas covered in Calculus 2 (Math 112) at the University of Science and Technology of Southern Philippines. It includes the following:
- Course outcomes focus on carrying out integration using fundamental formulas and techniques for single and multiple integrals.
- Topic outline covers anti-differentiation, simple power formulas, and simple trigonometric functions.
- Worked examples demonstrate evaluating indefinite integrals using power, trigonometric, and other basic integration rules.
- Important notes emphasize that the general solution for an indefinite integral includes an unknown constant C and the differential dx.
This document provides an introduction to integration (calculus) as taught in an undergraduate engineering course. It defines integration as the reverse process of differentiation and describes how it can be used to find the area under a curve. The document outlines key integration terminology like indefinite integrals, definite integrals, and the constant of integration. It also provides examples of integrating common functions and using integration to calculate volumes of solids of revolution.
This document provides a review of exercises for a Math 112 final exam. It contains 31 multi-part exercises covering topics like graphing, logarithms, trigonometry, and word problems. The review is intended to help students practice problems similar to what may appear on the exam. The exam will have two parts, one allowing a calculator and one not.
The document discusses transformations of linear functions. It provides examples of translating graphs vertically and horizontally by adding or subtracting values from inputs and outputs. It also discusses reflecting graphs over the x-axis or y-axis by multiplying inputs or outputs by -1. Horizontal and vertical stretches and shrinks are described as multiplying the inputs or outputs by a scale factor. The key is that transformations preserve the shape of the graph but can change its position, orientation, or scale.
Unit-1 Basic Concept of Algorithm.pptxssuser01e301
The document discusses various topics related to algorithms including algorithm design, real-life applications, analysis, and implementation. It specifically covers four algorithms - the taxi algorithm, rent-a-car algorithm, call-me algorithm, and bus algorithm - for getting from an airport to a house. It also provides examples of simple multiplication methods like the American, English, and Russian approaches as well as the divide and conquer method.
This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.
The document provides an overview of indefinite integration or anti-differentiation. It discusses standard integrals, rules of integration, techniques like substitution, integration by parts, partial fractions, and integrals of various functions. Some example integrals are also presented along with their solutions. Reduction formulas for integrals of trigonometric functions like secx and cosecx are outlined as well.
logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
Introduction:
[Start with a brief introduction about yourself, including your profession or main area of expertise.]
Background:
[Discuss your background, education, and any relevant experiences that have shaped your journey.]
Accomplishments:
[Highlight notable achievements, awards, or significant projects you've been involved in.]
Expertise:
[Detail your areas of expertise, skills, or specific knowledge that sets you apart in your field.]
Passions and Interests:
[Share your passions, hobbies, or interests outside of your professional life, adding depth to your personality.]
Vision or Mission:
[If applicable, articulate your vision, mission, or goals in your chosen field or in life in general.]
Closing Statement:
[End with a closing statement that summarizes your essence or leaves a lasting impression.]
Feel free to customize each section with your own personal details and experiences. If you need further assistance or have specific points you'd like to include, feel free to let me know!
Integral Calculus Anti Derivatives reviewerJoshuaAgcopra
This document provides an overview of integration concepts and formulas covered in Calculus 2 (Math 112) at the University of Science and Technology of Southern Philippines. It includes the following:
- Course outcomes focus on carrying out integration using fundamental formulas and techniques for single and multiple integrals.
- Topic outline covers anti-differentiation, simple power formulas, and simple trigonometric functions.
- Worked examples demonstrate evaluating indefinite integrals using power, trigonometric, and other basic integration rules.
- Important notes emphasize that the general solution for an indefinite integral includes an unknown constant C and the differential dx.
This document provides an introduction to integration (calculus) as taught in an undergraduate engineering course. It defines integration as the reverse process of differentiation and describes how it can be used to find the area under a curve. The document outlines key integration terminology like indefinite integrals, definite integrals, and the constant of integration. It also provides examples of integrating common functions and using integration to calculate volumes of solids of revolution.
This document provides a review of exercises for a Math 112 final exam. It contains 31 multi-part exercises covering topics like graphing, logarithms, trigonometry, and word problems. The review is intended to help students practice problems similar to what may appear on the exam. The exam will have two parts, one allowing a calculator and one not.
The document discusses transformations of linear functions. It provides examples of translating graphs vertically and horizontally by adding or subtracting values from inputs and outputs. It also discusses reflecting graphs over the x-axis or y-axis by multiplying inputs or outputs by -1. Horizontal and vertical stretches and shrinks are described as multiplying the inputs or outputs by a scale factor. The key is that transformations preserve the shape of the graph but can change its position, orientation, or scale.
Unit-1 Basic Concept of Algorithm.pptxssuser01e301
The document discusses various topics related to algorithms including algorithm design, real-life applications, analysis, and implementation. It specifically covers four algorithms - the taxi algorithm, rent-a-car algorithm, call-me algorithm, and bus algorithm - for getting from an airport to a house. It also provides examples of simple multiplication methods like the American, English, and Russian approaches as well as the divide and conquer method.
This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.
The document provides an overview of indefinite integration or anti-differentiation. It discusses standard integrals, rules of integration, techniques like substitution, integration by parts, partial fractions, and integrals of various functions. Some example integrals are also presented along with their solutions. Reduction formulas for integrals of trigonometric functions like secx and cosecx are outlined as well.
logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
Introduction:
[Start with a brief introduction about yourself, including your profession or main area of expertise.]
Background:
[Discuss your background, education, and any relevant experiences that have shaped your journey.]
Accomplishments:
[Highlight notable achievements, awards, or significant projects you've been involved in.]
Expertise:
[Detail your areas of expertise, skills, or specific knowledge that sets you apart in your field.]
Passions and Interests:
[Share your passions, hobbies, or interests outside of your professional life, adding depth to your personality.]
Vision or Mission:
[If applicable, articulate your vision, mission, or goals in your chosen field or in life in general.]
Closing Statement:
[End with a closing statement that summarizes your essence or leaves a lasting impression.]
Feel free to customize each section with your own personal details and experiences. If you need further assistance or have specific points you'd like to include, feel free to let me know!
This document contains notes from a calculus workshop covering several topics:
1) Arc length and applications of integrals.
2) Probability density functions and using integrals to find probabilities and means.
3) Parametric equations and eliminating parameters to sketch curves.
4) Vectors, dot products, cross products, and using them to find angles between vectors.
5) Coordinate systems including Cartesian, polar, cylindrical and spherical coordinates.
6) Double and triple integrals including finding areas, volumes, and changing coordinates.
This document discusses integration, which is the inverse operation of differentiation. It begins by explaining that integration finds the original function given its derivative, with the addition of a constant of integration. It then provides examples of basic integration techniques using a table of integrals. The document also outlines some rules for integrating sums and constant multiples of functions. Finally, it gives an example of using integration to solve an engineering problem involving the electric potential of a charged sphere.
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
This document provides instruction on operations of functions, including addition, subtraction, multiplication, and division. It begins by explaining how to add and subtract functions by applying the functions to the variable x. For addition, (f+g)(x) is defined as f(x) + g(x), and for subtraction, (f-g)(x) is defined as f(x) - g(x). It then covers multiplying functions by defining the product (f⋅g)(x) as f(x)⋅g(x) and dividing functions by defining the quotient (f/g)(x) as f(x)/g(x) where g(x) cannot be zero. Examples
This document discusses various matrix decomposition techniques including least squares, eigendecomposition, and singular value decomposition. It begins with an introduction to the importance of linear algebra and decompositions for applications. Then it provides examples of using least squares to fit curves to data and find regression lines. It defines eigenvalues and eigenvectors and provides examples of eigendecomposition. It also discusses diagonalization of matrices and using the eigendecomposition to raise matrices to powers. Finally, it discusses singular value decomposition and its applications.
This document discusses quadratic functions and how to find the vertex of a parabola. It explains that the vertex formula for a quadratic function f(x) = ax^2 + bx + c is x = -b/2a and y = f(-b/2a). Examples are provided of using the vertex formula or completing the square to change a quadratic function into vertex form and find the vertex. Students are assigned practice problems finding vertices through completing the square.
This document provides an overview of evaluating functions. It begins with defining a function and function notation. It then discusses the difference between solving a function and evaluating a function. The main steps for evaluating a function are outlined as: 1) write the original function, 2) substitute the given value for the variable, and 3) simplify using order of operations. Several examples are provided of evaluating different types of functions at given values by following these steps. The document concludes by restating the key steps for evaluating functions.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.
The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant base. It provides examples and discusses key characteristics of exponential graphs such as their domains, ranges, and asymptotic behavior. The document also covers transformations of exponential graphs and using exponential functions to model compound interest over time.
This document discusses using integration to find the area between two curves by considering it as an accumulation process. It explains that we select a representative element, such as a rectangle, and use geometry formulas to relate the area of that element to the functions that define the curves. The area under each representative rectangle is summed to find the total area via integration. Two examples are provided, one finding the area between a parabola and the x-axis using vertical rectangles, and another using horizontal rectangles between two other curves.
The document discusses various mathematical concepts related to functions and graphs including:
1) Transformations of graphs such as translations, reflections, and rotations. It also discusses parent functions and their derivatives.
2) Examples of graphing functions after applying transformations to translate, scale, or reflect the original graphs. Equations are provided for the transformed graphs.
3) Theorems related to how statistics of data change after translations or scale changes. For example, the mean, median and mode change proportionally but variance, standard deviation, and range change in specific ways.
4) Concepts involving inverse functions, including using the horizontal line test to determine if an inverse is a function and notations for inverse functions
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
I am Bella A. I am a Statistical Method In Economics Assignment Expert at economicshomeworkhelper.com/. I hold a Ph.D. in Economics. I have been helping students with their homework for the past 9 years. I solve assignments related to Economics Assignment.
Visit economicshomeworkhelper.com/ or email info@economicshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Method In Economics Assignments.
This document provides an overview of Mathematica, including information about its creator Stephen Wolfram, notational conventions, built-in mathematical constants, and keyboard shortcuts for entering symbols. It also includes examples evaluating mathematical expressions in Mathematica, such as verifying Safford's famous calculation of 133,491,850,208,566,925,016,658,299,941,583,225 as the square of 365,365,365,365,365,365 and calculating 100 digits of pi and e.
20230411 Discussion of Integration by Substitution.docxSharon Liu
1) The document discusses integration by substitution and evaluates some simple integrals using this method, such as ∫b(ax) dx and ∫(xa)b dx.
2) It then looks at calculating the integral of sin(3x) between 0 and 180 degrees by numerically evaluating the area under the curve in an Excel spreadsheet.
3) Some issues with using the standard table of integrals to evaluate ∫sin(x) dx from 0 to 360 degrees are noted, as it does not give the correct numeric result of 0.
This document discusses integrals involving exponential functions. It shows that integrating the exponential function results in dividing the constant in the exponent. It evaluates the important definite integral from 0 to infinity of e^-ax, which equals 1/a. It also evaluates the double integral from -infinity to infinity of e^-a(x^2+y^2), which equals sqrt(pi/a). Taking derivatives of these integrals generates related integrals involving x and x^4 that are useful in kinetic theory of gases.
1. The document discusses parametric equations, which express the variables x and y in terms of a third variable called a parameter. Common parameters include s, t, and θ.
2. It provides examples of converting parametric equations to Cartesian form by eliminating the parameter through substitution or trigonometric identities. This includes the equations of circles, parabolas, ellipses, and hyperbolas.
3. Key parametric equations that define common curves are identified, along with the curves they represent. Methods for sketching curves from their parametric equations are also outlined.
The document discusses various techniques for integration including integration by parts, trigonometric substitution, algebraic substitution, reciprocal substitution, and partial fraction decomposition. Integration by parts allows one to integrate products of functions. Trigonometric substitution transforms integrals into ones involving trigonometric functions that can be evaluated using basic formulas. Algebraic substitution rationalizes irrational integrals. Partial fraction decomposition expresses rational functions as sums of simpler fractions to facilitate integration.
Abhay Bhutada, the Managing Director of Poonawalla Fincorp Limited, is an accomplished leader with over 15 years of experience in commercial and retail lending. A Qualified Chartered Accountant, he has been pivotal in leveraging technology to enhance financial services. Starting his career at Bank of India, he later founded TAB Capital Limited and co-founded Poonawalla Finance Private Limited, emphasizing digital lending. Under his leadership, Poonawalla Fincorp achieved a 'AAA' credit rating, integrating acquisitions and emphasizing corporate governance. Actively involved in industry forums and CSR initiatives, Abhay has been recognized with awards like "Young Entrepreneur of India 2017" and "40 under 40 Most Influential Leader for 2020-21." Personally, he values mindfulness, enjoys gardening, yoga, and sees every day as an opportunity for growth and improvement.
This document contains notes from a calculus workshop covering several topics:
1) Arc length and applications of integrals.
2) Probability density functions and using integrals to find probabilities and means.
3) Parametric equations and eliminating parameters to sketch curves.
4) Vectors, dot products, cross products, and using them to find angles between vectors.
5) Coordinate systems including Cartesian, polar, cylindrical and spherical coordinates.
6) Double and triple integrals including finding areas, volumes, and changing coordinates.
This document discusses integration, which is the inverse operation of differentiation. It begins by explaining that integration finds the original function given its derivative, with the addition of a constant of integration. It then provides examples of basic integration techniques using a table of integrals. The document also outlines some rules for integrating sums and constant multiples of functions. Finally, it gives an example of using integration to solve an engineering problem involving the electric potential of a charged sphere.
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
This document provides instruction on operations of functions, including addition, subtraction, multiplication, and division. It begins by explaining how to add and subtract functions by applying the functions to the variable x. For addition, (f+g)(x) is defined as f(x) + g(x), and for subtraction, (f-g)(x) is defined as f(x) - g(x). It then covers multiplying functions by defining the product (f⋅g)(x) as f(x)⋅g(x) and dividing functions by defining the quotient (f/g)(x) as f(x)/g(x) where g(x) cannot be zero. Examples
This document discusses various matrix decomposition techniques including least squares, eigendecomposition, and singular value decomposition. It begins with an introduction to the importance of linear algebra and decompositions for applications. Then it provides examples of using least squares to fit curves to data and find regression lines. It defines eigenvalues and eigenvectors and provides examples of eigendecomposition. It also discusses diagonalization of matrices and using the eigendecomposition to raise matrices to powers. Finally, it discusses singular value decomposition and its applications.
This document discusses quadratic functions and how to find the vertex of a parabola. It explains that the vertex formula for a quadratic function f(x) = ax^2 + bx + c is x = -b/2a and y = f(-b/2a). Examples are provided of using the vertex formula or completing the square to change a quadratic function into vertex form and find the vertex. Students are assigned practice problems finding vertices through completing the square.
This document provides an overview of evaluating functions. It begins with defining a function and function notation. It then discusses the difference between solving a function and evaluating a function. The main steps for evaluating a function are outlined as: 1) write the original function, 2) substitute the given value for the variable, and 3) simplify using order of operations. Several examples are provided of evaluating different types of functions at given values by following these steps. The document concludes by restating the key steps for evaluating functions.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.
The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant base. It provides examples and discusses key characteristics of exponential graphs such as their domains, ranges, and asymptotic behavior. The document also covers transformations of exponential graphs and using exponential functions to model compound interest over time.
This document discusses using integration to find the area between two curves by considering it as an accumulation process. It explains that we select a representative element, such as a rectangle, and use geometry formulas to relate the area of that element to the functions that define the curves. The area under each representative rectangle is summed to find the total area via integration. Two examples are provided, one finding the area between a parabola and the x-axis using vertical rectangles, and another using horizontal rectangles between two other curves.
The document discusses various mathematical concepts related to functions and graphs including:
1) Transformations of graphs such as translations, reflections, and rotations. It also discusses parent functions and their derivatives.
2) Examples of graphing functions after applying transformations to translate, scale, or reflect the original graphs. Equations are provided for the transformed graphs.
3) Theorems related to how statistics of data change after translations or scale changes. For example, the mean, median and mode change proportionally but variance, standard deviation, and range change in specific ways.
4) Concepts involving inverse functions, including using the horizontal line test to determine if an inverse is a function and notations for inverse functions
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
I am Bella A. I am a Statistical Method In Economics Assignment Expert at economicshomeworkhelper.com/. I hold a Ph.D. in Economics. I have been helping students with their homework for the past 9 years. I solve assignments related to Economics Assignment.
Visit economicshomeworkhelper.com/ or email info@economicshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Method In Economics Assignments.
This document provides an overview of Mathematica, including information about its creator Stephen Wolfram, notational conventions, built-in mathematical constants, and keyboard shortcuts for entering symbols. It also includes examples evaluating mathematical expressions in Mathematica, such as verifying Safford's famous calculation of 133,491,850,208,566,925,016,658,299,941,583,225 as the square of 365,365,365,365,365,365 and calculating 100 digits of pi and e.
20230411 Discussion of Integration by Substitution.docxSharon Liu
1) The document discusses integration by substitution and evaluates some simple integrals using this method, such as ∫b(ax) dx and ∫(xa)b dx.
2) It then looks at calculating the integral of sin(3x) between 0 and 180 degrees by numerically evaluating the area under the curve in an Excel spreadsheet.
3) Some issues with using the standard table of integrals to evaluate ∫sin(x) dx from 0 to 360 degrees are noted, as it does not give the correct numeric result of 0.
This document discusses integrals involving exponential functions. It shows that integrating the exponential function results in dividing the constant in the exponent. It evaluates the important definite integral from 0 to infinity of e^-ax, which equals 1/a. It also evaluates the double integral from -infinity to infinity of e^-a(x^2+y^2), which equals sqrt(pi/a). Taking derivatives of these integrals generates related integrals involving x and x^4 that are useful in kinetic theory of gases.
1. The document discusses parametric equations, which express the variables x and y in terms of a third variable called a parameter. Common parameters include s, t, and θ.
2. It provides examples of converting parametric equations to Cartesian form by eliminating the parameter through substitution or trigonometric identities. This includes the equations of circles, parabolas, ellipses, and hyperbolas.
3. Key parametric equations that define common curves are identified, along with the curves they represent. Methods for sketching curves from their parametric equations are also outlined.
The document discusses various techniques for integration including integration by parts, trigonometric substitution, algebraic substitution, reciprocal substitution, and partial fraction decomposition. Integration by parts allows one to integrate products of functions. Trigonometric substitution transforms integrals into ones involving trigonometric functions that can be evaluated using basic formulas. Algebraic substitution rationalizes irrational integrals. Partial fraction decomposition expresses rational functions as sums of simpler fractions to facilitate integration.
Similar to technical-mathematics-integration-17-feb_2018.pptx (20)
Abhay Bhutada, the Managing Director of Poonawalla Fincorp Limited, is an accomplished leader with over 15 years of experience in commercial and retail lending. A Qualified Chartered Accountant, he has been pivotal in leveraging technology to enhance financial services. Starting his career at Bank of India, he later founded TAB Capital Limited and co-founded Poonawalla Finance Private Limited, emphasizing digital lending. Under his leadership, Poonawalla Fincorp achieved a 'AAA' credit rating, integrating acquisitions and emphasizing corporate governance. Actively involved in industry forums and CSR initiatives, Abhay has been recognized with awards like "Young Entrepreneur of India 2017" and "40 under 40 Most Influential Leader for 2020-21." Personally, he values mindfulness, enjoys gardening, yoga, and sees every day as an opportunity for growth and improvement.
Independent Study - College of Wooster Research (2023-2024) FDI, Culture, Glo...AntoniaOwensDetwiler
"Does Foreign Direct Investment Negatively Affect Preservation of Culture in the Global South? Case Studies in Thailand and Cambodia."
Do elements of globalization, such as Foreign Direct Investment (FDI), negatively affect the ability of countries in the Global South to preserve their culture? This research aims to answer this question by employing a cross-sectional comparative case study analysis utilizing methods of difference. Thailand and Cambodia are compared as they are in the same region and have a similar culture. The metric of difference between Thailand and Cambodia is their ability to preserve their culture. This ability is operationalized by their respective attitudes towards FDI; Thailand imposes stringent regulations and limitations on FDI while Cambodia does not hesitate to accept most FDI and imposes fewer limitations. The evidence from this study suggests that FDI from globally influential countries with high gross domestic products (GDPs) (e.g. China, U.S.) challenges the ability of countries with lower GDPs (e.g. Cambodia) to protect their culture. Furthermore, the ability, or lack thereof, of the receiving countries to protect their culture is amplified by the existence and implementation of restrictive FDI policies imposed by their governments.
My study abroad in Bali, Indonesia, inspired this research topic as I noticed how globalization is changing the culture of its people. I learned their language and way of life which helped me understand the beauty and importance of cultural preservation. I believe we could all benefit from learning new perspectives as they could help us ideate solutions to contemporary issues and empathize with others.
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"Does Foreign Direct Investment Negatively Affect Preservation of Culture in the Global South? Case Studies in Thailand and Cambodia."
Do elements of globalization, such as Foreign Direct Investment (FDI), negatively affect the ability of countries in the Global South to preserve their culture? This research aims to answer this question by employing a cross-sectional comparative case study analysis utilizing methods of difference. Thailand and Cambodia are compared as they are in the same region and have a similar culture. The metric of difference between Thailand and Cambodia is their ability to preserve their culture. This ability is operationalized by their respective attitudes towards FDI; Thailand imposes stringent regulations and limitations on FDI while Cambodia does not hesitate to accept most FDI and imposes fewer limitations. The evidence from this study suggests that FDI from globally influential countries with high gross domestic products (GDPs) (e.g. China, U.S.) challenges the ability of countries with lower GDPs (e.g. Cambodia) to protect their culture. Furthermore, the ability, or lack thereof, of the receiving countries to protect their culture is amplified by the existence and implementation of restrictive FDI policies imposed by their governments.
My study abroad in Bali, Indonesia, inspired this research topic as I noticed how globalization is changing the culture of its people. I learned their language and way of life which helped me understand the beauty and importance of cultural preservation. I believe we could all benefit from learning new perspectives as they could help us ideate solutions to contemporary issues and empathize with others.
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1. INTEGRAL CALCULUS (Integration)
Technical Mathematics
Grade 12-Just In Time Training
17 February 2018 ,
Ishaak Cassim
DCES: Technical Mathematics
Ishaak Cassim - February 2018 1
2. Outcomes for this section
• By the end of this section, you should be able to:
• Understand the concept of integration as a summation function (definite
integral) and as converse of differentiation (indefinite integral).
• Apply standard forms of integrals as a converse of differentiation.
• Integrate the following functions:
o kxn, with n ∈ ℝ; with 𝑛 ≠ – 1
o
𝒌
𝒙
and kanx with a > 0 ; k , a ∈ ℝ
• Integrate polynomials consisting of terms of the above forms.
• Apply integration to determine the magnitude of an area included by a
curve and the x-axis or by a curve, the x-axis and the ordinates x = a
and x = b where a , b ∈ ℤ.
Ishaak Cassim - February 2018 2
3. Introduction
The process of integration reverses the process of differentiation. In differentiation, if f(x) =
2x2
, then 𝑓′
(x) = 4x. Thus the integral of 4x is 2x2
. We can represent this process pictorially
as follows:
The situation gets a bit more complicated, because there are an infinite number of functions
we can differentiate to give 4x. Here are some of those functions:
f(x) = 2x2
+ 7 ; g(x) = 2x2
– 8 ; h(x) = 2x2
+
1
2
.
Ishaak Cassim - February 2018 3
4. Activity 1
• Activity 1
• Write down at least five other functions whose
derivative is 4x.
Ishaak Cassim - February 2018 4
5. Introduction
• You would have noticed that all the functions have the same
derivative of 4x, because when we differentiate the constant
term we obtain zero.
• Hence, when we reverse the process, we have no idea what
the original constant term might have been.
• So we include in our response an unknown constant, c, called
the arbitrary constant of integration.
• The integral of 4x then is 2x2 + c.
Ishaak Cassim - February 2018 5
6. Introduction
• In differentiation, the differential coefficient
𝑑𝑦
𝑑𝑥
indicates that
a function of x is being differentiated with respect to x, the dx
indicating that it is “with respect to x”.
• In integration, the variable of integration is shown by adding
d(the variable) after the function to be integrated. When we
want to integrate a function, we use a special notation:
𝒇 𝒙 𝒅𝒙.
• Thus, to integrate 4x, we will write it as follows:
Ishaak Cassim - February 2018 6
7. Introduction
4𝑥 𝒅𝒙 = 2x2
+ c , c ∈ ℝ.
Integral
sign
This term is called
the integrand
There must always be
a term of the form dx
Constant of
integration
Ishaak Cassim - February 2018 7
8. • Note that along with the integral sign ( 𝑑𝑥), there is a term of the form dx,
which must always be written, and which indicates the variable involved, in our
example x.
• We say that 4x is integrated with respect x, i.e: 𝟒𝒙 𝒅𝒙
• The function being integrated is called the integrand.
• Technically, integrals of this type are called indefinite integrals, to distinguish
them from definite integrals, which we will deal with later.
• When you are required to evaluate an indefinite integral, your answer must
always include a constant of integration.; i.e:
𝟒𝒙 𝒅𝒙 = 2𝑥2
+ c; where c ∈ ℝ
Ishaak Cassim - February 2018 8
9. Definition of anti-derivative
• Formally, we define the anti-derivative as: If f(x) is a
continuous function and F(x) is the function whose derivative
is f(x), i.e.: 𝑭′
(x) = f(x) , then:
𝒇 𝒙 𝒅𝒙 = F(x) + c; where c is any arbitrary
constant.
Ishaak Cassim - February 2018 9
10. Activity 2
1. In each of the following determine the function f(x), if the derivative 𝑭′
(x) is given:
No. 𝑭′
(x) f(x)
1. 4x
2. x4
3. 2x2
4. 0
5.
𝒙
𝟑
𝟐
6. -2x3
7. 𝟏
𝟓
𝒙𝟏𝟎
2. Explain how the derivative 𝑭′
(x) and the function f(x)are related to each other
Ishaak Cassim - February 2018 10
12. The general solution of integrals of the form kxn
• From Activity 2 above, the general solution of integrals of the
form 𝒌𝒙𝒏
dx , where k and n are constants is given by:
𝒌𝒙𝒏
dx =
𝒌𝒙𝒏+𝟏
𝒏+𝟏
+ c ; where 𝒏 ≠ −𝟏 and c ∈ ℝ
Ishaak Cassim - February 2018 12
13. Table of ready to use integrals
Function, f (x) Indefinite integral 𝒇(𝒙)𝒅𝒙
f (x)= k , where k is a constant 𝒌𝒅𝒙 = kx + c ; where c ∈ ℝ
f (x)= x 𝒙𝒅𝒙 =
𝟏
𝟐
𝒙𝟐
+ c; where c ∈ ℝ
f (x) = x2
𝒙𝟐
𝒅𝒙 =
𝟏
𝟑
𝒙𝟑
+ c; where c ∈ ℝ
f(x) = axn
𝒂𝒙𝒏𝒅𝒙 =
𝒂𝒙𝒏+𝟏
𝒏+𝟏
+ c; where c ∈ ℝ
f (x) = 𝒙−𝟏
=
𝟏
𝒙
𝟏
𝒙
dx = ln 𝒙 + c; where c ∈ ℝ
f (x) = kanx
𝒌𝒂𝒏𝒙
dx =
𝒌𝒂𝒏𝒙
𝒏.𝒍𝒏𝒂
+ c ; where c ∈ ℝ ; a > 0 and a ≠ 1
Ishaak Cassim - February 2018 13
14. Worked Examples: Indefinite integrals
a)
𝑥7
dx
=
𝒙𝟕+𝟏
𝟕+𝟏
+ c ; where c∈ ℝ
=
𝒙𝟖
𝟖
+ c ; where c∈ ℝ
Compare 𝑥7
dx with
𝒂𝒙𝒏
𝒅𝒙 =
𝒂𝒙𝒏+𝟏
𝒏+𝟏
+ c,
Then: a = 1; n = 7 ; n + 1 = 8
Ishaak Cassim - February 2018 14
15. Worked Examples: Indefinite integrals
c) 2
𝑢2du
= 2𝑢−2du
= 2 𝑢−2du
= 2 (
𝑢−2+1
−2+1
) + c ; where c∈ ℝ
=
−𝟐
𝒖
+ c ; where c∈ ℝ
Note: in this example, we
are integrating with
respect to u.
Explain the method used
Ishaak Cassim - February 2018 15
16. Worked Examples: Indefinite integrals
d) 𝑥 dx
= 𝒙
𝟏
𝟐 dx
=
=
𝟐
𝟑
+ c; where ………..
Complete
Ishaak Cassim - February 2018 16
17. Activity 3
1. Use the method outlined above to find a general expression for the function f(x) in each of the following cases.
𝑭′
(x) f(x)
a) 4x3
b) 6x5
c) 2x
d) 3x2 + 5x4
e) 10x9 – 8x7 – 1
f) –7x6 + 3x2 + 1
g) 1 – 3x-2
h) (x – 2)2 –
3
𝑥2
i) −2
5𝑥−1 +
3
5
𝑥
Ishaak Cassim - February 2018 17
18. Activty 3 - continued
2. Determine the indefinite integrals in the following cases
a) (9𝑥2 – 4𝑥 − 5)𝑑𝑥 b) (12𝑥2 + 6𝑥 + 4)𝑑𝑥
c) −5 𝑑𝑥 d) (16𝑥3 − 6𝑥2 + 10𝑥 − 3)𝑑𝑥
e) (2𝑥3
+ 5𝑥)𝑑𝑥 f) (𝑥 + 2𝑥2
)𝑑𝑥
g) (2𝑥2– 3x – 4)dx h) (1 − 2𝑥 − 3𝑥2)𝑑𝑥
3. Determine the following indefinite integrals
a) 1
𝑥3dx b) (𝑥2 –
1
𝑥2)dx
c) 𝑥 𝑑𝑥 d)
6𝑥
2
3 𝑑𝑥
e) 6𝑥4+5
𝑥2 dx
f) 1
𝑥
dx
Ishaak Cassim - February 2018 18
19. Mid-ordinate rule :Revision
Example 1: The area of an irregular metal plate needs to be calculated. The ordinates are drawn 5 cm apart across the surface of the
metal plate. The lengths of the ordinates in cm’s are: 29; 32; 33; 32,5; 32; 31; 31; 32; 33; 35; 37; 39; 40.
Solution:
Ordinate Calculation Mid-ordinate
29 𝟐𝟗 + 𝟑𝟐
𝟐
30,5
32 𝟑𝟐 + 𝟑𝟑
𝟐
32,5
33 𝟑𝟑 + 𝟑𝟐, 𝟓
𝟐
32,75
32,5 𝟑𝟐, 𝟓 + 𝟑𝟐
𝟐
32,25
32 𝟑𝟐 + 𝟑𝟏
𝟐
31,5
31 𝟑𝟏 + 𝟑𝟏
𝟐
31
31 𝟑𝟏 + 𝟑𝟐
𝟐
31,5
32 𝟑𝟐 + 𝟑𝟑
𝟐
32,5
33 𝟑𝟑 + 𝟑𝟓
𝟐
34
35 𝟑𝟓 + 𝟑𝟕
𝟐
36
37 𝟑𝟕 + 𝟑𝟗
𝟐
38
39 𝟑𝟗 + 𝟒𝟎
𝟐
39,5
40
Sum of mid-ordinates 402
Area of sheet
metal = k sum of
mid-ordinates
= 5 402
= 2010 cm2
Ishaak Cassim - February 2018 19
20. Area of irregular shapes- Revision
• We can verify, the accuracy of our work in example 1 above,
by using the following formula:
• Area = k × [(average of first & last ordinate)+ (sum of rest
of ordinates)]
= 5 × [(
29+40
2
) +( 32+33+32,5+32+31+31+32+33+35+37+39)]
= 5[ 34,5+ 367,5]
= 2010 cm2
Ishaak Cassim - February 2018 20
21. Revision: Area of irregular shapes
• Thus, when we are using the ordinates:
We use the mid-ordinate rule:
Sum of mid-ordinates = (Average of first and last ordinates) +
(Sum of rest of ordinates), and
Area = k × Sum of mid-ordinates, where k is the “width”
Ishaak Cassim - February 2018 21
22. Example 2: Area of irregular shape
Example 2: Determine the area of an irregular metal plate. Ordinates are drawn
1,5 cm apart. The lengths of the ordinates in cm are: 0;30;42;49;56;46;38;30;18;0
Solution:
Sum of mid-ordinates = (average of first and last ordinates) + (sum of other
ordinates)
= [(
0+0
2
) + (30 + 42+49+56+46+38+30+18)]
= [ 0 + 309]
= 309
Area = k × Sum of mid-ordinates
= 1,5 (309)
= 463,5 cm2
Ishaak Cassim - February 2018 22
23. The figure alongside is a trapezium with AB // CD and
FE perpendicular AD.
Area trapezium =
1
2
× (sum of parallel sides) × perp.
distance
=
1
2
× (AB + CD) × (AD)
But
1
2
× (AB + CD) = mid -ordinate FE …….(1)
∴ Area of trapezium = FE × AD…………..(2) E
A
B
C
D
F
Ishaak Cassim - February 2018 23
24. Introduction to definite integral
O
x
A
C
B
D
I
E
H
G
F
J
In the figure above, OABC is bounded by a base OC , two
vertical ordinates OA and BC and a curve AB.
Ishaak Cassim - February 2018 24
25. Introduction to definite intrgral
• To determine the area below the curve, divide the base OC into any
number of equal parts, each with length k units.
• Each of the strips, OADT, TDES, SEFR, RFGP, PGHN, NHIM, MIJL
and LJBC resemble trapezia (the plural of trapezium).
• Halfway, between each of the ordinates we draw mid-ordinates, i.e the
average of any two consecutive ordinates, as denoted by the dotted lines.
• We can thus calculate the area of each trapezium using the formula
developed above,
• Area of each strip = k × length of mid-ordinate, where k is the length of
the base of each trapezium
Ishaak Cassim - February 2018 25
26. Example 1
• Consider the following example: Plot the graph of y = 3x –x2,
by completing a table of values of y from x = 0 to x = 3.
• Determine the area enclosed by the curve, the x-axis and
ordinates x = 0 and x = 3 using the mid-ordinate rule.
x 0 0,5 1,0 1,5 2,0 2,5 3,0
y = 3x – x2 0 1,25 2 2,25 2 1,25 0
Ishaak Cassim - February 2018 26
27. Example 1
• Using the mid-ordinate rule with six intervals, where the mid-
ordinates are located at:
Area ≈ (0,5)[0,6875 + 1,6875 + 2,1875 + 2,1875 + 1,6875 +
0,6875]
= (0,5)(9,125)
= 4,563 square units
Mid -ordinate 0,25 0,75 1,25 1,75 2,25 2,75
Correspondin
g y-values
0,6875 1,6875 2,1875 2,1875 1,6875 0,6875
Ishaak Cassim - February 2018 27
28. Example 2: y = -x2 + 4
Area = 10 sq.
units
Ishaak Cassim - February 2018 28
29. Example 2
NB: As the
number of
partitions
increase the
area comes
closer to 9.
Ishaak Cassim - February 2018 29
30. Activity 4
1. Using rectangles, calculate the areas described in each of the following cases. Use a partition with a sensible number
of rectangles. Make sure each rectangle has the same width. Make a rough sketch for each case.
a) f(x) = 2x + 7 between the x-axis, and x = – 2 and x = 3.
b) k(x) = x2 bounded by the x –axis and x = – 4 and x = – 1.
c) j(x) =
𝟏
𝒙
, with x > 0, bounded by the x-axis and x = 1 and x = 4.
d) m(x) = x2 + 2, between the x-axis and x = – 5 and x = 3. Is this area an under- or overestimate of the true area?
e) p(x) = 25– 𝑥2 between the x-axis and the two x-intercepts. Can you work out the exact area under this curve? How
accurate was your partitioning method?
f) h(x) = 2 sinx, between the x-axis and x =
𝝅
𝟒
and x = 𝜋.
2. Create an Excel spreadsheet to approximate the area bounded by the curve
t(x) = –x2 + 4 on the interval [–1 ; 2] for 3 different rectangle widths.
3. Determine the area under the function g(x) =
–12
𝑥
, for x ∈ [– 1 ; 6], by dividing the given interval into five trapeziums.
Let the height of each trapezium be 1 unit. Use the following formula to assist you to calculate the area of each
trapezium:
Area =
𝟏
𝟐
× (sum of parallel sides) × height
3.1 How does this method compare with method of dividing the given interval into rectangles?
Ishaak Cassim - February 2018 30
31. What about areas that fall below the x-axis?
y = x(x-2)(x +2)
Ishaak Cassim - February 2018 31
32. Area below the x-axis
• Whether you use the rectangle or trapezium method to calculate the area bounded by the curve,
the x-axis, between x = –2 and x = 2, you would use all the y-values as positive lengths in your
calculations. Alternatively, the total area is double the area under the curve from x = – 2 to x = 0.
• Using the trapezium rule, with partitions of
1
2
unit wide, the area of the left-hand half under the
curve, between x = –2 and x = 0 is:
Area = sum of areas of the trapeziums
= {[
1
2
× [ f(–2) +f(-1,5)] ×
1
2
} + {[
1
2
× [ f(–1,5) +f(-1)] ×
1
2
} +…+{[
1
2
× [ f(–1) +f(0)] ×
1
2
}
=
1
4
[0 +2,625 + 2,625+ 3 + 3+ 1,875 + 1,875 + 0]
=
15
4
square units OR (3,75 square units)
• So, total area = 2 (
𝟏𝟓
𝟒
) =
𝟏𝟓
𝟐
square units OR (7,5 square units)
Ishaak Cassim - February 2018 32
33. The definite integral
• In the previous section we developed a method (a long method!) of finding the
area between the curve and the x-axis.
• The general notation for the area under the curve is 𝒂
𝒃
𝒇 𝒙 𝒅𝒙. This is known
as the definite integral of f(x).
• We define the definite integral of a function f(x) as :
𝒂
𝒃
𝒇 𝒙 𝒅𝒙 = F(b) – F(a); where 𝑭′
(x) = f(x).
• We call this a definite integral because the result of integrating and evaluating
is a number. (The indefinite integral has an arbitrary constant in the result).
• The numbers a and b are called the lower limit and upper limit, respectively.
• We can see that the value of a definite integral is found by evaluating the
function (found by integration) at the upper limit and subtracting the value of
this function at the lower limit.
Ishaak Cassim - February 2018 33
34. The definite integral
• The definite integral, 𝒂
𝒃
𝒇 𝒙 𝒅𝒙 can be interpreted as the
area under the curve of y = f(x) from x = a to x = b, and in
general as a summation.
• NB: If we asked to evaluate the definite integral, without
mention of calculating area, one would proceed as follows:
• Evaluate: −𝟐
𝟐
𝒙𝟑
𝒅𝒙 =
𝟏
𝟒
[𝟐𝟒
– (– 𝟐)𝟒
] = 0
Ishaak Cassim - February 2018 34
35. The fundamental Theorem of Calculus
• If f(x) is continuous on the interval 𝒂 ≤ 𝒙 ≤ 𝒃 and if F(x) is any
indefinite integral of f(x), then:
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = [𝐹 𝑥 ]𝑎
𝑏
= F(b) – F(a)
• This theorem is showing us how to evaluate an integral once the
integration process has taken place.
• NB: When evaluating the definite integral there is no need to
consider the arbitrary constant.
Ishaak Cassim - February 2018 35
36. Worked examples – the definite integral
Evaluate the following definite integrals
1
𝟐
𝟓
𝟖𝐱𝐝𝐱 Let I = 2
5
8𝑥𝑑𝑥
=
8
2
[𝑥2 ]2
5
using definition
= 4[52 – 22] using fundamental
theorem of calculus
= 4[ 25 - 4]
= 4[21]
∴ I = 84
Ishaak Cassim - February 2018 36
37. Worked examples – the definite integral
Evaluate the following definite integrals
2.
𝟎
𝟐
𝟔𝐱 + 𝟕 𝐝𝐱 Let I = 𝟎
𝟐
𝟔𝐱 + 𝟕 𝒅𝒙
= [𝟑𝒙𝟐
+ 𝟕𝐱]𝟎
𝟐
= [3 (2)2 + 7(2)] – [3(0)2 + 7(0)]
= 12 + 14 – 0
∴ I = 26
Ishaak Cassim - February 2018 37
38. Worked examples- The definite integral
3.
𝟎
𝟏
𝟐𝒅𝒙 Let I = 0
1
2𝑑𝑥
= [2𝑥]0
1
= [2(1) – 2(0)]
∴ I = 2
Ishaak Cassim - February 2018 38
39. Worked examples- The definite integral
4.
𝟐
𝟒 𝟏𝟐
𝒙
dx Let I = 2
4 12
𝑥
dx
= 12 2
4 1
𝑥
dx
= 12 𝒍𝒏 𝑥 ]2
4
= 12(ln 4 – ln 2)
∴ I = 12ln 2
Ishaak Cassim - February 2018 39
40. Activity 5
1. Evaluate the following definite integrals, using the method outlined above.
a)
−1
1
𝑥 − 1 (𝑥 + b)
0
1
2𝑥 − 3 𝑑𝑥
c)
0
1
3𝑥 + 2 𝑑𝑥 d)
–1
0
5𝑥4
dx
e)
1
2 𝑑𝑥
𝑥2
f)
1
2 𝑥4+1
𝑥2 dx
g)
2
3
10𝑛𝑥
dx h)
−2
3
(𝑥2
+ 5)𝑑𝑥
Ishaak Cassim - February 2018 40
41. Some properties of integrals
1.
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = – 𝑏
𝑎
𝑓 𝑥 𝑑𝑥
2.
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 + 𝑏
𝑐
𝑓 𝑥 𝑑𝑥 = 𝑎
𝑐
𝑓 𝑥 𝑑𝑥
3 Constants may be factored through the integral sign:
𝒂
𝒃
𝒌. 𝒇 𝒙 𝒅𝒙 = k. 𝒂
𝒃
𝒇 𝒙 𝒅𝒙
4. The integral of a sum (and /or difference) is the sum (and /or difference) of the integrals:
𝒂
𝒃
[𝒇 𝒙 ± 𝒈 𝒙 ]𝒅𝒙 = 𝒂
𝒃
𝒇 𝒙 𝒅𝒙 ± 𝒂
𝒃
𝒈 𝒙 𝒅𝒙
5. The integral of a linear combination is the linear combination of the integrals:
𝒂
𝒃
[𝒌. 𝒇 𝒙 + 𝒎. 𝒈 𝒙 ]𝒅𝒙 = k. 𝒂
𝒃
𝒇 𝒙 𝒅𝒙 + m. 𝒂
𝒃
𝒈 𝒙 𝒅𝒙
Here are some simple properties of the integral that are often used in computations. Throughout take f
and g as continuous functions.
Ishaak Cassim - February 2018 41
42. Activity 6
Property 3: 𝒂
𝒃
𝒌. 𝒇 𝒙 𝒅𝒙 = 𝒌. 𝒂
𝒃
𝒇 𝒙 𝒅𝒙
• With your partner, explore the validity of property 3 using
the following functions:
o f(x) = x2 and k = 2
o f(x) = 2x and k = –0,5
o f(x) =
𝟏
𝒙𝟐 and k = –1,5
• Comment on your findings.
Ishaak Cassim - February 2018 42
43. Activity 7
Investigate the validity of the following property:
• 𝒂
𝒃
𝒇 𝒙 + 𝒈 𝒙 𝒅𝒙 = 𝒂
𝒃
𝒇(𝒙)𝒅𝒙 + 𝒂
𝒃
𝒈(𝒙)𝒅𝒙,
if f(x) = –3x and g(x) = x2 + 6 for the interval [–4 ;–1].
o You will need to draw graphs of the two functions on the same set
of axis.
o You will need to show all working details.
• Comment on your findings.
Ishaak Cassim - February 2018 43
45. Area under a curve
• Use the properties of definite integrals developed in the
previous section to calculate the area under a curve.
• NB: If asked to evaluate a definite integral without explicitly
asked to calculate the area, then proceed to evaluate the
definite integral without concern about the answer-i.e the
answer can be 0 or a negative value!!
Ishaak Cassim - February 2018 45
46. Area under a curve- Worked example
• Find the area under the curve y = x2 , between x = 1 and x =
3.
Let the required
area be A , then:
A = 1
3
𝑥2𝑑𝑥
= [
𝑥3
3
]1
3
=
33
3
–
13
3
A =
𝟐𝟔
𝟑
square
units
Ishaak Cassim - February 2018 46
47. Area under a curve
Find the area between the curve y = x2 + 4x and the
x-axis from:
ox = – 2 to x = 0
ox = 0 to x = 2
ox = – 2 to x = 2
Ishaak Cassim - February 2018 47
A1
A2
48. Worked example 2
For the interval [– 2 ; 2], we observe that the required area comprises of two parts – one part below the x-axis and the other part
above the x-axis.
Let A1 = −𝟐
𝟎
𝒙𝟐
+ 𝟒𝒙 𝒅𝒙
= [
𝑥3
3
+ 2𝑥2
]−2
0
= 0 –[
−23
3
+ 2(−2)2
]
= 0 – [
16
3
]
= –
16
3
…..the minus sign tells us the area is below the x-axis
Thus, A1 =
𝟏𝟔
𝟑
• Calculate A2 , as follows:
A2 = 0
2
𝑥2
+ 4𝑥 𝑑𝑥 =
32
3
(The reader should verify the accuracy)
• So, the area over the interval [–2 ; 2] is obtained by adding A1 and A2, i.e:
A = A1 + A2
=
𝟏𝟔
𝟑
+
𝟑𝟐
𝟑
= 16 square units.
Ishaak Cassim - February 2018 48
49. Own work
With your partner, attempt the following.
The given sketch is a graphical representation of the function
defined by: f(x) = x3 – 4x2 + 3x.
Determine the area between the curve and the x-axis from x = 0
to x = 3.
Ishaak Cassim - February 2018 49
50. Challenge
• Find the area enclosed between the curve
y = x2 – 2x –3 and the straight-line y = x + 1.
Ishaak Cassim - February 2018 50
51. Activity 9
1. Use the properties of definite integrals which you have studied thus far,
to evaluate the following definite integrals.
a)
−2
2
6𝑡2
+ 1 𝑑𝑡 b)
0
1
2𝑥 + 1 𝑥 + 3 𝑑𝑥
c)
2
4
5𝑥 − 4 𝑑𝑥 d)
−3
3
6𝑥3 + 2𝑥 𝑑𝑥
2. Calculate the areas of the following:
a)
−2
2
6𝑡2
+ 1 𝑑𝑡 b)
0
1
2𝑥 + 1 𝑥 + 3 𝑑𝑥
c)
2
4
5𝑥 − 4 𝑑𝑥 d)
−3
3
6𝑥3
+ 2𝑥 𝑑𝑥
Ishaak Cassim - February 2018 51
52. Activity 9
3. Find the area under the curve y = x2 from x = 0 to x = 6.
4. Find the area under the curve y = 3x2 + 2x from x = 0 to x = 4.
5. Find the area under the curve y = 3x2 -2x from x = – 4 to x = 0.
6. Find the area enclosed by the x-axis and the following curves and straight
lines.
a) y = x2 + 3x ; x = 2 ; x =5 b) y =
1
8
x3 + 2x ; x = 2 ; x =4
c) y = (3x–4)2 ; x = 1 , x =3 d) y = 2 – x3 ; x = –3 ; x = – 2
7.a) Sketch the curve y = x(x +1)(x –3), showing where it cuts the x-axis.
b) Calculate area of the region, above the x-axis, bounded by the x-axis and the
curve.
c) Calculate area of the region, below the x-axis, bounded by the x-axis and the
curve. Ishaak Cassim - February 2018 52
53. Activity 9
8. The diagram shows the region under
y = 4x +1 between x =1 and x = 3. Find
the area below the graph between x = 1
and x= 3, using:
a) The formula for the area of a trapezium
b) integration
c) How do the results in a) and b) compare
with each other? Explain.
9. The diagram alongside shows the region
bounded by y =
1
2
𝑥 – 3, by x =14 and the
x-axis. A(r ; 0) is the x-intercept of the
straight line graph.
a) Determine the numerical value of r.
b) Determine area of the shaded region
using:
i) the formula for the area of a triangle.
ii) integration
1 3
y = 4x +1
A (r ;0) 14
y =
1
2
𝑥 – 3
x
Ishaak Cassim - February 2018 53
54. Activity 9
10. Calculate the areas of the following:
a) y = 1 + x2 for
0 < x < 2
b) y = 5x – x2 ; for x ∈ [0 ; 5]
11. Evaluate 0
2
𝑥 𝑥– 1 𝑥– 2 𝑑𝑥 and explain your answer
with reference to the graph of
y = x(x–1)(x–2).
12. Given that: −𝑎
𝑎
15𝑥2
dx = 3430. Determine the
numerical value of a
13. Given that: 1
𝑝
(8𝑥3
+ 6𝑥)𝑑𝑥 = 39. Determine two
possible values of p. Use a graph to explain why there
are two values.
14. Show that the area enclosed between the curves
y = 9 –x2 and y = x2 – 7 is
128 2
3
Ishaak Cassim - February 2018 54