Nigeria's free digital learning hub. Videos, lecture notes, practice questions and leaderboards aligned with CCMAS, NBTE, NCCE, WAEC, NECO, IJMB and JAMB curricula.
Loading
Nigeria's free digital learning hub. Videos, lecture notes, practice questions and leaderboards aligned with CCMAS, NBTE, NCCE, WAEC, NECO, IJMB and JAMB curricula.
Mathematics · 300 Level
This is the complete lecture note for MTH 307 Real Analysis II, prepared for Mathematics students at 300 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1: Riemann Integral of Functions in Rn Series Outline Part 1.1: Definition and Properties of the Riemann Integral Sub-Part 1.1.1: Basic definition of the Riemann integral Sub-Part 1.1.2: Properties of integrable functions Part 1.2: Riemann Integrability in Rn Sub-Part 1.2.1: Extension of the Riemann integral to higher dimensions Sub-Part 1.2.2: Examples of integrable functions in Rn Part 1.3: Continuous and Monopositive Functions Sub-Part 1.3.1: Definition of continuous functions in Rn Sub-Part 1.3.2: Monopositive functions and their integrability Part 1.4: Applications of the Riemann Integral Sub-Part 1.4.1: Evaluation of definite integrals in Rn Sub-Part 1.4.2: Practical examples and problem sets Introduction The study of integration is central to real analysis, and the Riemann integral provides the foundation for understanding how functions can be accumulated over intervals. Extending this concept to functions defined on Rnallows us to handle multidimensional problems, which are essential in mathematics, physics, and engineering. This Series will guide you through the definition, properties, and applications of the Riemann integral in higher dimensions, preparing you for more advanced topics such as bounded variation and Stieltjes integration. The aim of this Series is to introduce you to the Riemann integral of functions in Rn, to explain its properties, and to demonstrate how it applies to continuous and monopositive functions. We shall commence this Series by examining the definition and properties of the Riemann integral. Next, we will extend the concept to functions in Rn, supported with illustrative examples. Following that, we will consider continuous and monopositive functions, highlighting their integrability. Finally, we will conclude
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS with applications of the Riemann integral, where you will see how definite integrals in higher dimensions can be evaluated and applied to practical problems. Series Learning Outcomes Upon completing this Series, you should be able to: SLO 1.1 define the Riemann integral and explain the fundamental properties of integrable functions, SLO 1.2 demonstrate how the Riemann integral extends to functions in Rnwith illustrative examples, SLO 1.3 describe continuous functions and define monopositive functions in Rn, analysing their integrability, SLO 1.4 apply the Riemann integral to evaluate definite integrals in higher dimensions and solve practical problems. 1.1 Definition and Properties of the Riemann Integral The Riemann integral is a fundamental concept in real analysis. It provides a way of measuring the total accumulation of a function’s values over an interval. The basic idea is to partition the interval into smaller subintervals, evaluate the function at chosen points within each subinterval, multiply these values by the lengths of the subintervals, and then sum the results. This sum is called a Riemann sum, and the Riemann integral is defined as the limit of these sums as the partitions become finer. Fill in the blank: The Riemann integral is defined as the limit of __________ sums as the partitions become finer. 1.1.1 Basic definition of the Riemann integral A function fdefined on a closed interval bis said to be Riemann integrable if the limit of its Riemann sums exists and is finite. Formally, if we partition binto subintervals and denote the maximum length of these subintervals as the mesh, then as the mesh approaches zero, the Riemann sums converge to a single value. This value is the Riemann integral of fover b.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Partitioning an interval for Riemann sums Fill in the blank: A function is Riemann integrable if the limit of its __________ sums exists and is finite. 1.1.2 Properties of integrable functions The Riemann integral has several important properties: Linearity: If fand gare integrable, then af+bgis also integrable, and ab(af(x)+bg(x))dx=aabf(x)dx+babg(x)dx
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Additivity: If fis integrable on b, then for any cin b, abf(x)dx=acf(x)dx+cbf(x)dx Monotonicity: If f(x)≤g(x)for all xin b, then abf(x)dx≤abg(x)dx Boundedness: A function must be bounded on bto be Riemann integrable. Properties of the Riemann integral Content: • Linearity: Integrals respect addition and scalar multiplication. • Additivity: Integrals can be split across subintervals. • Monotonicity: Larger functions yield larger integrals. • Boundedness: Functions must be bounded to be integrable
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: For a function to be Riemann integrable, it must be __________ on the interval. Pilot Questions 1.1 The Riemann integral is defined as the limit of: A. Derivatives B. Riemann sums C. Infinite series D. None of the above (Linked to SLO 1.1) A function is Riemann integrable if the limit of its sums: A. Does not exist B. Exists and is finite C. Is infinite D. None of the above (Linked to SLO 1.1) Which property states that integrals can be split across subintervals? A. Linearity B. Additivity C. Monotonicity D. Boundedness (Linked to SLO 1.1) If f(x)≤g(x)for all xin b, then: A. abf(x)dx≥abg(x)dxB. abf(x)dx≤abg(x)dxC. abf(x)dx=abg(x)dxD. None of the above (Linked to SLO 1.1) For a function to be Riemann integrable, it must be: A. Continuous everywhere B. Bounded on the interval C. Differentiable everywhere D. None of the above (Linked to SLO 1.1) 1.2 Riemann Integrability in Rn The concept of the Riemann integral can be extended beyond single-variable functions to functions defined on higher-dimensional spaces such as R2or R3. In these cases, instead of partitioning an interval, we partition regions such as rectangles or cubes into smaller subregions. The function values are then multiplied by the volumes of these subregions, and the sums are taken. The limit of these sums, as the partitions become finer, defines the Riemann integral in higher dimensions. Fill in the blank: In higher dimensions, the Riemann integral is defined by partitioning regions such as __________ or cubes into smaller subregions. 1.2.1 Extension of the Riemann integral to higher dimensions
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS For a function f(x,y)defined on a rectangle a,b]×[c,d, the Riemann integral is obtained by dividing the rectangle into smaller rectangles. Each rectangle contributes an area multiplied by the function value at a chosen point. As the mesh of the partition approaches zero, the sum converges to the double integral: abcdf(x,y) dy dx This idea generalises to Rn, where the integral is defined over n-dimensional regions. Partitioning a rectangle in R2for Riemann sums
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: The double integral of a function f(x,y)over a rectangle is written as \int_a^b \int_c^d __________ \, dy \, dx. 1.2.2 Examples of integrable functions in Rn Consider the function f(x,y)=x+ydefined on the unit square 0,1]×[0,1. The Riemann integral is: 0101(x+y) dy dx Evaluating this gives: 01xy+y22∣01dx=01x+12dx=12+12=1 Thus, the integral of f(x,y)=x+yover the unit square is 1. [Insert Plate here] Short description: Visual representation of the function f(x,y)=x+yover the unit square. Caption: Plate 1.1 Graph of f(x,y)=x+yover the unit square AI prompt: “Generate a 3D surface plot of the function f(x,y) = x + y over the unit square [0,1] × [0,1].” Search string: “surface plot x+y unit square OER” Fill in the blank: The integral of f(x,y)=x+yover the unit square 0,1]×[0,1is __________. Pilot Questions 1.2 In higher dimensions, the Riemann integral is defined by partitioning: A. Intervals B. Rectangles or cubes C. Circles D. None of the above (Linked to SLO 1.2) The double integral of a function f(x,y)over a rectangle is written as: A. abf(x)dxB. abcdf(x,y)dydxC. cdf(y)dyD. None of the above (Linked to SLO 1.2) The function f(x,y)=x+yover the unit square integrates to: A. 0 B. 1 C. 2 D. None of the above (Linked to SLO 1.2) Partitioning a rectangle into smaller rectangles for integration is an example of: A. Riemann sums in higher dimensions B. Taylor series expansion C. Fourier analysis D. None of the above (Linked to SLO 1.2)
This is the opening of the note. Read the full note or download the PDF to continue.
