What is Riemann sum: Definition and 76 Discussions

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.
The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution.
Because the region filled by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the shapes get smaller and smaller, the sum approaches the Riemann integral.

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  1. chwala

    Integration problem using inscribed rectangles

    Just went through this...steps pretty clear. I refreshed on Riemann integrals { sum of rectangles approximate area under curves}. My question is on the highlighted part in Red. The approximation of area under curve may be smaller or larger than the actual value. Thus the inequality may be ##<##...
  2. crememars

    Finding a definite integral from the Riemann sum

    Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...
  3. crememars

    Identifying variables from Riemann sum limits

    Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...
  4. Graham87

    Numerical Analysis - Richardson Extrapolation on Riemann Sum

    I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?). For example sin(x) for [0,1], n with trapezoidal rule = 0.420735... With my own...
  5. S

    Riemann Sum to find the time to fill a container

    (a) I imagine there are several rectangles to represent the area under graph of p vs t then I try to make equation for the total area. Since the question asks about time when the container holds 22 fewer liters than it does at time t = 9, I think the total area of rectangles starting from t = b...
  6. Leo Liu

    Midpoint Riemann sum approximation

    Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like $$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$? Thanks in advance!
  7. karush

    MHB 2.7.3 AP calculus Exam Riemann sum

    ok basically t is 3 hours appart except between 7 and 12 of which I didn't know if we should intemperate. other wise it is just adding up the 4 $(t)\cdot(R(t))$s.
  8. PainterGuy

    MATLAB Approximating the inverse FT of a unit pulse using a Riemann sum

    Hi, Although I'm using trigonometric form of Fourier transform, first I'd discuss both, exponential and trigonometric forms, for the sake of context. Now proceeding toward the main question and we would only be using trigonometric form. % file name...
  9. D

    How to prove that ##f(x,y)## is not integrable over a square?

    I'm confused with how Riemann sums work on double integrals. I know that ##L=\sum_{i,j}fm_{ij}A_{ij}## and ##U=\sum_{i,j}fM_{ij}A_{ij}## where ##m_{ij}## is the greatest lower bound and ##M_{ij}## is the least uper bound and ##A_{ij}## is the area of each partition. ##A_{ij}=\frac{1}{n^2}## for...
  10. Zack K

    Calculating area using sigma notation

    Homework Statement The question involves using sigma notation of Riemann sums to find the area under the graph of ##x^2+\sqrt {1+2x}##. I managed to calculate most of the values and I have ##16+\frac 8 3 + \Sigma {\frac 2 n \sqrt {9 + \frac {4i} n}}## Homework Equations [/B] ##\Sigma i= \frac...
  11. M

    MHB Volume Problem - Riemann Sum

    THE QUESTION By using Riemann’s sum, synthesise a mathematical model for finding the exact volume of any ‘tepee’ tent of side s and height h. HERE'S WHAT I HAVE Am currently stuck on writing a side length for the hexagon at any height 'x'
  12. D

    Calculating the Limit Using Riemann Sum with Starred Part?

    Homework Statement http://i66.tinypic.com/aesd1u.png can someone explain to me how can i get the limit using riemann sum especially the starred part? i was so confused thanks! Homework Equations The Attempt at a Solution attempt at a solution in the picture
  13. D

    Find the limit using Riemann sum

    Homework Statement i want to find limit value using riemann sum \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br> question : <br> \lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br> Homework EquationsThe Attempt at a...
  14. FallArk

    MHB How to calculate the lower and upper riemann sum

    I ran into some issues when trying to calculate the lower Riemann sum of f\left(x\right)={x}^{3}, x\in[0,1] I am asked to use the standard partition {P}_{n} of [0,1] with n equal subintervals and evaluate L(f,{P}_{n}) and U(f,{P}_{n}) What I did: L(f,{P}_{n}) =...
  15. TheSodesa

    Number of subdivisions in a Riemann integral (DFT)

    Homework Statement This is a combination of two questions, one being the continuation of the other 3) Calculate the DFT of the sequence of measurements \begin{equation*} \{ g \}_{k=0}^{5} = \{ 1,0,4,-1,0,0 \} \end{equation*} 4a) Draw the DFT calculated in question 3 on the complex plane. 4b)...
  16. MAGNIBORO

    I Solving Riemann Sum Problem: Integral of x^x

    Hi. I try to solve the integral $$\int_{0}^{1} x^{x} dx$$ Through sums of riemann But I came to the conclusion that the result is 0 that is wrong $$\int_{0}^{1} x^{x} dx = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}$$ $$= \lim_{n\rightarrow...
  17. Rectifier

    Understanding the Riemann Sum - Integral Connection

    The problem I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty## The attempt ## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow...
  18. F

    I Proofs of various integral properties

    I've been trying to prove a couple of properties of integrals using the Riemann sum definition: $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned (such that ##a=x_{1}<x_{2}<\cdots <x_{i-1}<x_{i}<\cdots...
  19. F

    I Definitions of the Riemann integral

    In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...
  20. T

    MHB Find a formula for the Riemann sum and take the limit of the sum as n->infinite

    For the function given below find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then take a limit of thissum as n-> infinite to calculate the area under the curve over [1,5]. Below you...
  21. S

    M: Solve Riemann Sum Problem Homework

    Homework Statement [/B] Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.Homework Equations f(x) = x2+ 1, over the interval [0,1]. This is problem number such-and-such from...
  22. S

    Solving Riemann Sum for "Deformation of Water by Magnetic Field

    I'm not sure if this is the right place to post this in, but I'm trying to recreate the "Deformation of water by a magnetic field" experiment by Chen et al. The PDF version of the paper can be accessed via Google (for some reason it won't let me provide a direct link). On the 2nd page of the...
  23. A

    MHB Approximating Position with Riemann Sums

    The question provides a table of values for time and velocity. Part c of the question asks to use a Riemann sum to approximate (not specifying which one). Part d asks what the answer to part c represents and to explain my reasoning. The answer that I got for the sum is 58.5 feet, but I do not...
  24. A

    MHB Finding an Integral for a given Riemann Sum

    Hello, This question is purely inspired by: http://mathhelpboards.com/calculus-10/evaluating-infinite-sum-e-x-using-integrals-12838.html My other question. Anyhow, How do you find the integral for a given specific Riemann sum. Suppose the same one given in the link; $= \displaystyle...
  25. A

    Trying to find the infinite sum of e^-x using integration

    Hello, I am well aware of the ratio method, and the sum = 1/(1-r) but I want to try this method. I am trying to understand this: \displaystyle \sum_{n=1}^{\infty} e^{-n} using integrals, what I have though: = \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n} = \displaystyle...
  26. P

    Lower and Upper Riemann sums of sin(x)

    Task in real analysis: P is a uniform partition on [0, π] and is divided into 6 equal subintervals. Show that the lower and upper riemann sums of sin (x) over P is lesser than 1.5 and larger than 2.4 respectively. My attempt at the solution: The greates value and the least value of sin x over...
  27. S

    Compute ∫√(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum

    Homework Statement Integrate √(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum Homework Equations lim n→∞ Σ_(i=1)^n i = n(n+1)/2 lim n→∞ Σ_(i=1)^n i^2 = n(n+1)(2n+1)/6 The Attempt at a Solution Δx = (b - a)/n Δx = (5 - 0)/n Δx = 5/n f(x_i) = √(25 - [a + iΔx]^2) f(x_i) = √(25 - [0 +...
  28. NATURE.M

    Proving the Riemann Sum for the Integral of x^2 from 1 to 3

    So my textbook asks to show \int^{3}_{1} x^{2}dx = \frac{26}{3}. They let the partition P = {x_{0},...,x_{n}}, and define the upper Riemann sum as U(P) = \sum^{i=1}_{n} x_{i}Δx_{i} and lower sum as L(P) = \sum^{i=1}_{n} x_{i-1}Δx_{i} I understand this part, but the next part is where I'm...
  29. I

    MHB Riemann Sum Definite Integral Question

    So the question is Evaluate (x-2)dx as the integral goes from -2 to 2 using the definition of a definite integral, choosing your sample points to be the right endpoints of the subintervals… Ok, so i understand how to do this problem if it gave me an actual number of interval like n=6 but it...
  30. MarkFL

    MHB Realguy's question at Yahoo Answers regarding a Riemann sum

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  31. Q

    What is the Riemann Sum Approximation for this Homework Problem?

    Homework Statement https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1456973_10201043975243279_1765184125_n.jpg?oh=05b39611ad70d28d837ed219e1b0f2aa&oe=52838593 Homework Equations The area can be approximated by using the sum of the areas of the rectangles. Area of rectangle = change...
  32. S

    Riemann sum where n = 3 for both left and right endpoint estimates

    Homework Statement Compute the integral that is highlighted in MyWork.jpg using Riemann sums using left and right endpoints. Homework Equations ##x_i* = a + i Δx## ##*x_i = a + i Δx - Δx## ##Σ_{i=1}^{n} i = n(n+1)/2## ##Σ_{i=1}^{n} i^2 = n(n+1)(2n+1)/6## The Attempt at a Solution My...
  33. MarkFL

    MHB Maya's question at Yahoo Answers regarding a Riemann sum and definite integral

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  34. Fernando Revilla

    MHB Solving an Integral with a Right Endpoint Riemann Sum

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  35. L

    Express e^x from 1 to 8 as a Riemann Sum. Please, check my work?

    Express e^x from 1 to 8 as a Riemann Sum. Please, check my work? :) 1. Express ∫1 to 8 of e^xdx as a limit of a Riemann Sum. (Please ignore the __ behind the n's. The format is not kept without it...) _____n 2. lim Ʃ f(xi)(Δx)dx x→∞ i=1 Δx= (b-a)/n = 8-1/n = 7/n xi= 1 + 7i/n ____n lim Ʃ...
  36. M

    Riemann Sum with subintervals/partition

    So I missed a class and am trying to figure out a question in my textbook but am completely lost. It goes a little something like this: Let f(x)=x3 and let P=<-2,0,1,3,4> be a partition of [-2,4]. a) Compute Riemann Sum S(f,P*) if the points <x1*,x2*,x3*,x4*>=<-1,1,2,4> are embedded in P...
  37. F

    Relationship between Riemann Sum and the Integral

    Homework Statement The notation for a Riemann sum - Ʃ f(x*i)Δx - is very similar to the notation for the integral (the Ʃ becomes ∫, the f(x*i) becomes f(x) and the Δx becomes dx). \int f(x)dx = \lim_{n \to \infty}\sum_{k=0}^{n} f(x_i) Δx Is there a way to explicitly define the values on the...
  38. R

    Proving Bounds for a Riemann Sum: Part II

    this is a riemann sum question and i need help with part 2 let Sn denote the finite sum 1+2^ 3/2 +...+n^ 3/2 i) use suitable upper and lower riemann sums for the function f(x)=x^3/2 on the interval [0,100] to prove that S99<J<100 ummm i did this and found 40000<J<41000 II) hence, or...
  39. joe_cool2

    Solve Riemann Sum Question for Area Under Curve

    *SOLVED*Riemann Sum Question *SOLVED* My question is quite simple. I probably just missed something somewhere. I've looked for hours and cannot find the mistake. Homework Statement Find the area under the curve using the definition of an integral and Gauss summation equations: f(x) = 3 -...
  40. H

    MHB Riemann sum to estimate volume

    Question: A solid has a rectangular base that lies in the first quadrant and is bounded by the x and y-axes and the lines x=2, y=1. The height of the solid above point (x,y) is 1+3x. Find the Riemann approximation of the solid. Solution: I already know that the solution is \sum_{i=1}^{n}...
  41. T

    What is the Integral Being Approximated by the Given Riemann Sum?

    Homework Statement For which integral, is the below example, a Riemann sum approximation.? The example is: 1/30( sqrt(1/30) + sqrt(2/30) + sqrt(3/30)+...+sqrt(30/30)) A. Integral 0 to 1 sqrt(x/30) B. Integral 0 to 1 sqrt(x) C. (1/30) Integral 0 to 30 sqrt(x) D. (1/30) Integral 0 to 1...
  42. F

    Riemann Sum Calculation for f(x)=x on [0,2] with n=8

    Let Pn denote the partition of the given interval [a,b] into n sub intervals of equal length Δxi = (b-a)/n Evaluate L(f,Pn) and U(f,Pn) for the given functions f and the given values of n. f(x)=x on [0,2], with n=8 2.My solution x0 = 0, x1 = 1/4, x2 = 1/2, x3 = 3/4, x4 = 1, x5 = 5/4...
  43. P

    Prove Riemann Sum: (ex-1)/x for x > 0

    Homework Statement Prove that: lim n->inf1/n*Ʃn-1k=0ekx/n = (ex-1)/x x>0 Homework Equations That was all the information provided. Any progress i have made is below. I didn't want to add any of that to this section because this is all speculation on my part so far. The Attempt at a...
  44. M

    Comp Sci Need fortran help Trapazoid riemann sum

    Alright, I cannot seem to get this subroutine to return the correct sums for the trapezoidal rule... Where do I need to fix? SUBROUTINE atrap(i) USE space_data IMPLICIT NONE INTEGER :: i, j REAL :: f_b1, f_b2, f_x1, f_x2, trap_area REAL :: delta_x trap_area = 0 f_b1 = lower f_b2...
  45. O

    Riemann Sum from Indefinite Integral

    Homework Statement Consider the integral, \int _3 ^7 (\frac{3}{x} + 2) dx a) Find the Riemann Sum for this integral using right endpoints and n=4. b) Find the Riemann Sum for this integral using left endpoints and n=4. Homework Equations The sum, \sum^{n = 4} (\frac{3}{x} + 2) The graph...
  46. J

    I really need some help with these Riemann sum problems

    Homework Statement 1. Express as a sum of riemann and write the integral to express the area of the trapezoid with vertex (0,0) , (1,3) , (3,3) , (5,0). 2. find the intersection points limited by these equations y = xsquare -3x and y = -2x +3 = 0 3. the trapezoid with vertex...
  47. QuarkCharmer

    Find Area Under Curve y=x^3 from 0 to 1: Riemann Sum Limit

    Homework Statement a.) Use definition 2 to find an expression for the area under the curve y=x^3 from 0 to 1 as a limit. b.)Evaluate the (above) limit using the sum of the cubes of the n integers. Homework Equations (\frac{n(n+1)}{2})^{2} The Attempt at a Solution For part a.) I wrote my...
  48. A

    Riemann sum -> Integral?

    Homework Statement Rn=\sum(i*e^(-2i/n))/n^2, i=1 Identify this Riemann sum corresponding to a certain definite integral. Homework Equations The Attempt at a Solution I got till 1/n^2 [1/e^(2/n)+2/e^(4/n)+3/e^(6/n)...n/e^2] and that's it. To my understanding I should be...
  49. T

    Math Real Analysis Problem, Riemann Sum Integral?

    Part 1. Homework Statement The problem literally states... " The Integral. limit of n-> infinity of n*[1/(n+1)^2 + 1/(n+2)^2 + 1/(n+3)^2 + 1/(2n)^2] = 1/2 " According to the teacher, the answer is 1/2. I don't know why or how to get there. Part 2. The attempt at a solution...
  50. A

    How can I convert the Riemann Sum into an Integral?

    If I have a function c(x,Δx) that gives the area between x and x + Δx of a function. The area under the function can be given by: Sum from j = 0 to n-1 of c(b/n j,c/b) As n tends to infinity and b is the upper limit of integration. How can I convert this from a sum into a integral? I'm not...
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