Mathematics
Riemann Sum
A Riemann sum is a method for approximating the area under a curve by dividing the region into narrow rectangles and summing their areas. It is a fundamental concept in calculus and is used to understand the behavior of functions and solve problems related to integration. Riemann sums are essential for developing the definite integral.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "Riemann Sum"
eBook - PDF- Neil A Watson(Author)
- 1993(Publication Date)
- WSPC(Publisher)
Chapter 7 THE RIEMANN INTEGRAL Definition of the integral Let f be a bounded, real-valued function on [a, 6]. We shall give a precise meaning to the idea of the signed area between the graph {(*,y): 2/= /(*), a 0 and negative where f(x) < 0. In calculus, your intuitive idea of area was used frequently, but you only have a rigorous definition of area for a few simple shapes such as rectangles. Our method involves approximation by rectangles, since we know their areas. DEFINITIONS. A finite set of numbers P = { x o , x i , . . . , x n _ i , x n } such that a = XQ < x < ... < x n -i < x n = 6 , is called a partition of [a, 6]. Each point x r is called a point of division, and each interval [ x r _ i , x r ] , for r € {1,...,n}, is called a subinterval formed by P. For each r , put M r = sup / , m r = inf / ; [£ r _l,S r ] fcr-l|£r] then M r (x r — x r _ i ) is the area of a rectangle which, intuitively, is greater than or equal to the area between the graph {(s,y): y = / ( x ) , X r -i < X< X R } and the x-axis, while mr(x r — x r -i) is the area of another rectangle which is less than or equal to that area. (Draw a picture!). Note that M r and T ot may be positive or negative, thus giving the rectangular areas a sign. Now put n n S(f, P) = M r (x r ~ X r -i) , *(/, P ) = Y1 m r( x r ~ a ?r-l) 5 r=l r=l these numbers are called the upper sum and lower sum of / for the partition P , respectively. The upper sum gives an upper estimate for the area we 130 Definition of the integral 131 seek, while the lower sum gives a lower estimate, the idea now is to take the infimum of the set of all upper sums for all possible partitions, and the supremum of all the lower sums, and define the area we seek to be their common value, if they are equal; they may not be. To do this, we must first show that the set of upper sums is bounded below, and that the set of lower sums is bounded above.
eBook - PDFWelcome to Real Analysis
Continuity and Calculus, Distance and Dynamics
- Benjamin B. Kennedy(Author)
- 2022(Publication Date)
- American Mathematical Society(Publisher)
7 The Riemann Integral In calculus, we learn that an integral computes the “area under the graph of a function” (more accurately, the “signed” area between the graph of the function and the horizon-tal axis). Somewhat more abstractly, we learn that the integral of a function ? over the interval [?, ?] is in some sense the “total” of ? as its inputs range from ? to ? , and that the integral divided by the length of the interval accordingly gives the “average” of ? on the interval. These interpretations of the integral are crucial in many applications. Our focus here, however, will be more on the remarkable fact that integration is es-sentially an inverse process for differentiation. This fact is what allows us to actually compute many integrals easily. Furthermore, it will be via the integral that we will be able to gain insight into a number of facts that are ostensibly about differentiation, including the existence of solutions of differential equations and the differentiation of power series. 7.1 Partitions and the definition of the integral In this section, we define the integral and begin to explore its properties. Throughout this chapter, we will be considering functions that are not necessarily continuous. We will, however, insist that the functions we consider be bounded . Here is the definition. Definition 7.1.1. Let ? be any set with ? ⊆ ? , and let ? ∶ ? → ℝ be a function. We say that ? is bounded on ? if there is some ? ≥ 0 such that ?(?) ∈ [−?, ?] for all ? ∈ ? . If ? = ? , we simply say that ? is bounded . Note that saying that ? is bounded on ? is the same as saying that ?(?) is a bounded set (recall Definition 5.1.6). As we develop the theory of partitions, we will need the following (by now) ele-mentary proposition. 235 236 Chapter 7. The Riemann Integral Proposition 7.1.2. Suppose that ? and ? are nonempty bounded subsets of ℝ such that ? ≤ ? for any ? ∈ ? and any ? ∈ ? .
eBook - PDF- William J. Terrell(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
Chapter 6 The Riemann Integral This chapter contains the most fundamental properties of the Riemann integral and consequences of the fundamental theorem of calculus. The primary geometric interpretation of the definite integral in introductory calculus is the computation of the area between the graph of a continuous function and its domain axis. Later in this book we extend this area measure of a planar region to a larger class of sets. 6.1. Partitions and Riemann-Darboux Sums The Riemann integral is defined for bounded functions on closed intervals. We begin with notation for finite collections of points that subdivide an interval [ a, b ] into subintervals. Definition 6.1.1. Let x 0 , x 1 , . . . , x n be points in the interval [ a, b ] with a = x 0 < x 1 < · · · < x n − 1 < x n = b . Then the set P = { x 0 , x 1 , . . . , x n } is called a partition of [ a, b ] . The subintervals of P are the intervals [ x k − 1 , x k ] for k = 1 , . . . , n . A partition P is a refinement of P if P ⊆ P . The functions we consider are bounded on [ a, b ] but not necessarily continuous at every point. So we must consider the supremum and infimum (rather than the maximum and minimum) of the function on each subinterval of a partition when defining our approximating sums, which are called Riemann-Darboux sums. Definition 6.1.2. Let f : [ a, b ] → R be bounded and let P = { x 0 , x 1 , . . . , x n } be a partition of [ a, b ] . Let M k = sup x ∈ [ x k − 1 ,x k ] f ( x ) and m k = inf x ∈ [ x k − 1 ,x k ] f ( x ) . The upper and lower Riemann-Darboux sums for f over [ a, b ] with parti-tion P are defined, respectively, by U ( f, P ) = n k =1 M k ( x k − x k − 1 ) and L ( f, P ) = n k =1 m k ( x k − x k − 1 ) . 149 150 6. The Riemann Integral For simplicity, the notation for upper and lower sums does not include any reference to the domain interval since the domain is fixed in most of the following discussion.
eBook - PDFTheories of Integration
The Integrals of Riemann, Lebesgue, Henstock–Kurzweil, and McShane
- Douglas S Kurtz, Charles W Swartz;;;(Authors)
- 2011(Publication Date)
- WSPC(Publisher)
Chapter 2 Riemann integral 2.1 Riemann’s definition The Riemann integral, defined in 1854 (see [Ri1], [Ri2]), was the first of the modern theories of integration and enjoys many of the desirable proper-ties of an integration theory. While the most popular integral discussed in introductory analysis texts, the Riemann integral does have serious short-comings which motivated mathematicians to seek more general integration theories to overcome them, as we will see in subsequent chapters. The groundwork for the Riemann integral of a function f over the in-terval [ a,b ] begins with dividing the interval into smaller subintervals. Definition 2.1 Let [ a,b ] ⊂ R . A partition of [ a,b ] is a finite set of numbers P = { x 0 ,x 1 ,...,x n } such that x 0 = a , x n = b and x i -1 < x i for i = 1 ,...,n . For each subinterval [ x i -1 ,x i ], define its length to be lscript ([ x i -1 ,x 1 ]) = x i -x i -1 . The mesh of the partition is then the length of the largest subinterval, [ x i -1 ,x i ]: μ ( P ) = max { x i -x i -1 : i = 1 ,...,n } . Thus, the points { x 0 ,x 1 ,...,x n } form an increasing sequence of numbers in [ a,b ] that divides the interval [ a,b ] into contiguous subintervals. Let f : [ a,b ] → R , P = { x 0 ,x 1 ,...,x n } be a partition of [ a,b ], and t i ∈ [ x i -1 ,x i ] for each i . As noted in Chapter 1, Riemann began by considering the approximating (Riemann) sums S ( f, P , { t i } n i =1 ) = n summationdisplay i =1 f ( t i ) ( x i -x i -1 ) , defined with respect to the partition P and the set of sampling points 11 12 Theories of Integration { t i } n i =1 . Riemann considered the integral of f over [ a,b ] to be a “limit” of the sums S ( f, P , { t i } n i =1 ) in the following sense.
eBook - PDFExploring Calculus
Labs and Projects with Mathematica
- Crista Arangala, Karen A. Yokley(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
Exercise: e. Since we have 4 coefficients, we must have 4 points in order to determine the coefficients. Plug in n = 1 into n summationdisplay i =1 i 2 = an 3 + bn 2 + cn + d to determine one equation of a, b, c, and d . f. Plug in n = 2 , 3 , and 4 into n summationdisplay i =1 i 2 = an 3 + bn 2 + cn + d to get three more equations. g. Use the Solve command to solve for a, b, c, and d . Do not forget that the Solve command requires == in each equation instead of a single equals. h. Plug a, b, c, and d back into n summationdisplay i =1 i 2 = an 3 + bn 2 + cn + d and factor in order to determine the general equation. i. Use the same technique to determine a general equation for n summationdisplay i =1 i 3 = an 4 + bn 3 + cn 2 + dn + e. Areas, Integrals, and Accumulation 65 Lab 20: Riemann Sums Introduction In mathematics, a Riemann Sum is an approximation of the area of a region, often the region underneath a curve and above the x -axis. It is named after German mathematician Bernhard Riemann. We will see how to do some com-mon Riemann Sums in this lab. Use the demonstration: http://demonstrations.wolfram.com/RiemannSums/ for the following exercises. FIGURE 3.1: A Visualization of a Riemann Sum Exercises: Set the number of rectangles to 4 and the function to x 2 − 1. a. Choose the left height (this will actually give the left sum, L 4 ). Determine the width and height of each of the rectangles. Use this information to determine the approximate area between the curve and the x -axis using the areas of the rectangles. (Note that if a height is negative, the “area” of this rectangle contributes a negative amount to the total area.) b. Choose right height (this will actually give the right sum, R 4 ). Determine the width and height of each of the rectangles. Use this information to 66 Exploring Calculus: Labs and Projects with Mathematica determine the approximate area between the curve and the x -axis using the areas of the rectangles.
eBook - PDF- Cesar E. Silva(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
Chapter 6 Integration 6.1. The Riemann Integral In this section we develop the theory of Riemann integration. The theory is devel-oped for functions defined on closed bounded intervals. In Section 6.3 we discuss the extension of the Riemann integral to unbounded intervals, which leads us to consider what are called improper Riemann integrals. For concreteness, let us first consider a function f : [ a, b ] → R that is positive. We are interested in defining the notion of the area under the graph of the function f that is over the interval [ a, b ], i.e., the area of the region { ( x, y ) : a ≤ x ≤ b, 0 ≤ y ≤ f ( x ) } . For example, if f ( x ) = 2, then this region is a rectangle and its area is 2( b − a ). But, for example, in the case of f ( x ) = x 2 , we do not have a readily available formula for the area. In this and more general cases our technique will be to approximate the region by rectangles and to use the sum of the areas of the rectangles as an approximation for the area of the region. The base of each rectangle will be a subinterval of [ a, b ], and the height will be given by the value of the function f at a point in this subinterval. To construct the subintervals of [ a, b ], we introduce the notion of a partition. Definition 6.1.1. A partition of an interval [ a, b ] consists of a finite sequence of points P = { x 0 , . . . , x n } such that a = x 0 < x 1 < · · · < x n = b. ♦ A partition P = { x 0 , . . . , x n } of [ a, b ] with n + 1 points gives a partition of the interval into the following n subintervals [ x 0 , x 1 ] , [ x 1 , x 2 ] , . . . , [ x n − 1 , x n ] . 197 198 6. Integration The i th subinterval of the partition is [ x i − 1 , x i ] , where i ranges from 1 to n , so the first subinterval ( i = 1) is [ x 0 , x 1 ]. In applications it is often convenient to use equally spaced points. Definition 6.1.2. The uniform partition of [ a, b ] consists of points { x i : i = 0 , .
- Lorenzo Sadun(Author)
- 2023(Publication Date)
- American Mathematical Society(Publisher)
(7.32) Do you notice anything about the derivatives of the accumulation functions that we have found so far? We’ll have a lot more to say about them in the next chapter. 7.5. Chapter Summary The Main Ideas. • Integration means adding up the pieces to make a whole. It applies to any quantity that can be broken into pieces, not just to the area under a curve. • Our strategy is to break the quantity into pieces, estimate the size of each piece, and add up the pieces. The whole is the sum of the parts. 7.5. Chapter Summary 209 • Most of the time, adding up the pieces results in an expression of the form ∑ =1 ( ∗ )Δ . This is called a Riemann Sum and can be visualized as the area under the curve = (). • Setting up Riemann Sums involves choices. We need to decide on the number of pieces, on the sizes of the intervals , and the choices of the representative points ∗ . • Some choices give much more accurate approximations than others. Both the trapezoidal rule and the midpoint rule are much more accurate than using left endpoints or right endpoints. • A definite integral is the limit of a Riemann Sum as the number of pieces goes to ∞. That is, (7.33) ∫ () = lim →∞ ∑ =1 ( ∗ )Δ . • If () is integrable, then all choices give the same limit as → ∞. (That’s the definition of integrability!) All continuous functions, and most other functions we will encounter, are integrable. • While integrals come from a variety of problems, all integrals can be visualized as signed areas. This allows us to apply geometric arguments to integrals. • The accumulation function () is the running total of the integral of (), namely ∫ 0 (). We often write () = ∫ 0 () , where is a dummy vari- able. So far we only know how to compute a few accumulation functions, but they will eventually turn out to be very useful. • If () is the accumulation function of (), then ∫ () = () − ().
eBook - PDF- Lorenzo Sadun(Author)
- 2023(Publication Date)
- American Mathematical Society(Publisher)
• While integrals come from a variety of problems, all integrals can be visualized as signed areas. This allows us to apply geometric arguments to integrals. • The accumulation function () is the running total of the integral of (), namely ∫ 0 (). We often write () = ∫ 0 () , where is a dummy vari- able. So far we only know how to compute a few accumulation functions, but they will eventually turn out to be very useful. • If () is the accumulation function of (), then ∫ () = () − (). Expectations. You should be able to • Apply the strategy of breaking things into pieces, estimating the pieces, and add- ing up the pieces to approximate a variety of bulk quantities as Riemann Sums. • Understand and use ∑ notation for sums. • Approximate bulk quantities numerically via Riemann Sums, using either left endpoints, right endpoints, midpoints, or the trapezoidal rule. • Take the limit as → ∞ to describe bulk quantities exactly as integrals. • Convert back and forth between a Riemann Sum and an integral that it approxi- mates. • Relate integrals to areas under, over, and between curves. • Understand and use the main properties of integrals. • Use accumulation functions to compute integrals of any linear function () = 0 + 1 . 7.6. Exercises Note: The exercises for this chapter are about understanding bulk quantities by slicing and dicing them into pieces that we can estimate, adding up the pieces, and taking a limit to get an integral. Some exercises are about setting up those integrals, some are about evaluating Riemann Sums numerically, and some are about accumulation functions. A few are about evaluating integrals that can be visualized as the area under a curve, but mostly the point is to understand how any bulk quantity, not just area, can be understood as an integral. 222 7. The Whole Is the Sum of the Parts Riemann Sums 7.1.
eBook - PDFThe Integral
A Crux for Analysis
- Steven G. Krantz, Steven Krantz(Authors)
- 2022(Publication Date)
- Springer(Publisher)
Since this is true for any > 0, we obtain the desired conclusion. ✷ 2.5 NUMERICAL TECHNIQUES OF INTEGRATION 2.5.1 INTRODUCTION Many of the most important integrals in mathematics, such as erf (N) = N −∞ e −x 2 dx (which represents the Gaussian normal distribution of probability theory), cannot be evaluated by standard methods of calculus. The reason is that it is impossible to write down an antiderivative for e −x 2 with a closed formula involving functions that we know (see [RIT] for a characterization of those functions that have closed form integrals). Nevertheless, it is crucial to be able to calculate such integrals to any desired degree of accuracy. The purpose of this section is to acquaint you with some techniques for doing so. 2.5.2 THE METHOD OF RECTANGLES Recall that the definition of the integral was as a limit of Riemann Sums. Let us agree that, throughout this section, we will only use Riemann Sums over uniform partitions (partitions having intervals of equal length): x j = x = (b − a)/k for all j. We will also take ξ j ∈ I j = [x j −1 , x j ] to be the right endpoint x j . Then b a f (x) dx = lim k→∞ k j =1 f (x j ) · x where, as the limit is being taken, the mesh of the partition {x 1 , ..., x k }, x = x j − x j −1 = b − a k , tends to zero. Refer to Figure 2.4. By the definition of limit, the value of the Riemann Sum is close to the value of the integral when the mesh is small. Again referring to Figure 2.4, we interpret this statement to mean that the value of the integral is closely approximated by the sum of the areas of the rectangles in the figure. 2.5. NUMERICAL TECHNIQUES OF INTEGRATION 17 Figure 2.4: Approximation of the integral by a Riemann Sum. We summarize all these ideas: Theorem 2.12 (The Method of Rectangles) Let f be a continuous function whose domain contains the interval [a, b]. The value of the integral b a f (x) dx is approximated, using the uniform partition {x 0 , . . . , x k }, by f (x 1 ) x + f (x 2 ) x + · · · + f (x k ) x.
eBook - PDF- Hugo D. Junghenn(Author)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 5 Riemann Integration on R 5.1 The Riemann–Darboux Integral Throughout this section, f denotes an arbitrary bounded, real-valued function on a closed and bounded interval [ a, b ] . The first step in the development of the Riemann–Darboux integral is to partition the interval [ a, b ] into finitely many subintervals, which are used to form upper and lower sums of f . Under suitable conditions, the sums converge to the integral. 5.1.1 Definition. A partition of [ a, b ] is a set P = { x 0 , x 1 , . . . , x n -1 , x n } , where x 0 := a < x 1 < · · · < x n -1 < x n := b. The points x 1 , . . . , x n -1 are called the interior points of the partition. The mesh of the partition is defined as P := max 1 ≤ j ≤ n Δ x j , where Δ x j := x j -x j -1 , 1 ≤ j ≤ n. A refinement of P is a partition containing P . The common refinement of partitions P and Q is the partition P ∪ Q . ♦ 5.1.2 Example. Let p ∈ N . Then, for each n ∈ N , P n := { j/p n : j = 0 , 1 , . . . , p n } is a partition of [0 , 1], P n = p -n , and P n +1 is a refinement of P n . ♦ 5.1.3 Definition. The lower and upper ( Darboux ) sums of f over a partition P of [ a, b ] are defined, respectively, by S ( f, P ) := n j =1 m j Δ x j and S ( f, P ) := n j =1 M j Δ x j , where m j = m j ( f ) := inf x j -1 ≤ x ≤ x j f ( x ) and M j = M j ( f ) := sup x j -1 ≤ x ≤ x j f ( x ) . ♦ 107 108 A Course in Real Analysis A geometric interpretation of the upper and lower sums for a positive continuous function is given in Figure 5.1. The lower (upper) sum is the total area of the smaller (larger) rectangles. a x 1 x 2 x 3 x 4 b x f FIGURE 5.1 : Upper and lower sums of f . The following proposition asserts that refinements increase lower sums and decrease upper sums. 5.1.4 Proposition. If Q is a refinement of P , then S ( f, P ) ≤ S ( f, Q ) ≤ S ( f, Q ) ≤ S ( f, P ) .
- Patrick Jones(Author)
- 2022(Publication Date)
- For Dummies(Publisher)
To produce the function, replace each expression of the form 5 i n that appears in the summation with the variable x . Notice that you can rewrite the summation as lim lim n i n i n n n i n i n n i n i n 1 3 3 1 3 5 5 125 5 5 5 ; that means you can replace 5 5 3 i n i n with x x 3 so that ( ) f x x x 3 . Therefore, the Riemann Sum could represent the definite integral 0 5 3 x x dx . 541. 6 To begin, you split the interval into n pieces of equal width using the formula x b a n , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have x n n 2 0 2 You also want to select a point from each interval. The formula x a x i i ( ) gives you the right endpoint from each interval. Here, you have x a x i i n i n i ( ) 0 2 2 Substituting those values into the definition of the integral gives you the following: 0 2 1 1 1 2 1 2 2 2 ( ) lim lim x dx f x x i n n n i i n i n n lim lim n i n i i n n n n i n n n i 1 1 1 2 1 4 2 1 4 lim ( ) lim ( ) n n n n n n n n n n 2 4 1 2 2 4 2 4 1 6 2 2 Note that formulas i n k kn 1 and ( ) i n i n n 1 1 2 were used to simplify the summations. 344 PART 2 The Answers ANSWERS 501–600 542. 196 To begin, you split the interval into n pieces of equal width using the formula x b a n , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have x n n 4 0 4 You also want to select a point from each interval. The formula x a x i i ( ) gives you the right endpoint from each interval. Here, you have x a x i i n i n i ( ) 0 4 4 Substituting those values into the definition of the integral gives you the following: 0 4 3 1 1 3 1 3 1 3 4 x dx f x x i n n i i n i n n lim lim 4 4 192 1 4 192 1 2 3 1 3 1 3 n n n i n n n n n i i n n n lim lim ( ) 2 2 2 4 192 1 4 192 1 4 19 n n n n n lim ( ) ( ) 6 Note that the formulas i n k kn 1 and i n i n n 1 3 2 1 2 ( ) were used to simplify the summations.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
