What is Riemann sums: Definition and 52 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. V

    Calculating Riemann Sums on Python w/ Numpy

    import numpy as np def num_int(f,a,b,n): dx=(b-a)/n x=np.arange(a,b,step=dx) y=f(x) return y.sum()*dx def rational_func(x): return 1/(1+x**2) print(num_int(rational_func,2,5,10)) Here is my code for the left endpoint, I know this code works because I compared it to an...
  2. S

    Riemann sums for discontinuous functions

    The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f##...
  3. P

    Calculus 2 for Engineers: Riemann sums

    Homework Statement a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1. b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4. The Attempt at a Solution I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...
  4. T

    MHB Definite integral with Riemann sums

    Heya, So, I know this is a pretty simple problem, but I seem stuck on it nevertheless. Here's the question Calculate the upper and lower sums , on a regular partition of the intervals, for the following integrals \begin{align*} \int_{1}^{3}(1-7x)dx \end{align*} Please correct me if I'm doing...
  5. MountEvariste

    MHB Sum of powers limit via Riemann sums?

    One of the many excellent problems by lfdahl in the challenge questions and puzzles subforum was recently: https://mathhelpboards.com/challenge-questions-puzzles-28/prove-limit-23480.html My first idea was Riemann sums! I didn't succeed. So I ask, can this limit be calculated via Riemann...
  6. P

    MHB Dale's questions via Facebook about Riemann Sums

    (a) Since a = 2, that means $\displaystyle \begin{align*} \Delta x = \frac{2}{5} \end{align*}$ (b) $\displaystyle \begin{align*} f \left( \frac{7\,\Delta x}{2} \right) &= f \left( \frac{7}{5} \right) \\ &= \left( \frac{7}{5} \right) ^2 \\ &= \frac{49}{25} \end{align*}$ (c) $\displaystyle...
  7. terryds

    Definite integral as Riemann sums

    Homework Statement Determine ##\int_{0}^{2}\sqrt{x}dx## using left riemann sums Homework Equations ##\int_{a}^{b}f(x)dx = \lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}(\frac{b-a}{n})f(x_i)## The Attempt at a Solution [/B] ##\frac{b-a}{n}=\frac{2-0}{n}=\frac{2}{n}## ##\int_{0}^{2}\sqrt{x}dx =...
  8. Nemo1

    MHB Solving Limits and Riemann Sums: Tips from Nemo

    Hi Community, I have the following question: I have done basic solving of limits and also of Riemann sums but never had to do them in the same question. Would I be correct in saying that I need to solve for the Riemann sum first then take the limit of the integral? Cheers Nemo
  9. N

    Upper and lower bound Riemann sums

    Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...
  10. N

    What are the definitions and properties of Riemann sums?

    Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...
  11. V

    MHB Riemann sums: linear integrand

    ∫(4−7x)dx ====> Integral is from 1 to 8 Similar question but for some reason I can't get the answer after following all the steps
  12. 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...
  13. 1

    Casio Program Help for AP Calc Riemann Sums

    Hi brand new to the site. I keep on having a syntax error when I run the code below on my casio fx-cg10. Btw I also put a display triangle on the last m as well
  14. Math Amateur

    MHB Computing a Simple Integral Via Riemann Sums

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Exercise 2(a) from Stoll's Exercises 6.2 on page 229 ... Exercise 2 reads as follows: I was somewhat puzzled about how to do this exercise ... BUT ... even more puzzled when I read Stoll's hint for solving the...
  15. 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...
  16. A

    MHB Basic Calculus II Integral Questions - Riemann Sums, Absolute Integrals, etc.

    Hey guys, I'd appreciate some help for this problem set I'm working on currently The u-substitution for the first one is somewhat tricky. I ended up getting 1/40(u)^5/2 - 2 (u) ^3/2 +C, which I'm not too sure about. I took u from radical 3+2x^4. For the second question, I split the integral...
  17. A

    Determination of limit in Riemann sums.

    How could I find the lim as n-> infinity of the expression I attached? The only way I could find was to express it in terms of a definite integral. Integral of xe^(-2x) from 0 to 1. What is the other way?
  18. H

    Finding limit of sum using Riemann Sums

    Find the limit limn→∞∑i=1 i/n^2+i^2 by expressing it as a definite integral of an appropriate function via Riemann sums ...?
  19. A

    How Do You Estimate a Double Integral Using Riemann Sums?

    Homework Statement If R = [0,4]x[-1,2], use a Riemann sum with m=2, n=3 to estimate the value of ∫∫(1-xy^2)dA. Take the sample points to be the lower right corners. Homework Equations NoneThe Attempt at a Solution 2*1[f(2,-1) + f(2,0) + f(2,1) + f(4,-1) + f(4,0) + f(4,1)] = some value Just...
  20. L

    ∫ of (x^2) from 0 to 2 proof w/o using fundamental therom, but w/ Riemann Sums

    Homework Statement \int_0^2 x^2 \, dx using true definition involving Riemann Sums (w/o fundamental theorem). Homework Equations I don't know what the relevant equations may be, perhaps some type of lim\sum f(x)(x_{j}-x_{j-1}) The Attempt at a Solution No attempt. Just seeking the...
  21. B

    Understanding Inf & Sup in Riemann Sums

    I am currently reading about riemann sums and several different sources uses these abbreviations, inf and sup, and I am not certain what they mean. Could someone explain them to me?
  22. M

    Matlab - multiple integral Riemann sums

    1. Homework Statement Write an m.file that will integrate a function f(x, y) on any given rectangle (a,b)\times(c,d) and returns the value of the integral from a to b and c to d of the function f(x,y) . Include error-catching code in the case that the integral diverges. The program...
  23. I

    Proving Integrability of Bounded Functions using Riemann Sums - Homework Help

    Homework Statement Suppose f:[a,b] → ℜ is bounded and for each ε > 0, ∃ a partition P such that for any refinements Q1 and Q2 of P, regardless of how marked ⎟S(Q1,f) – S(Q2,f)⎟ < ε. Prove that f is integrable on [a,b]. Homework Equations If P and Q1 and Q2 are partitions of [a, b], with...
  24. V

    Theory behind Integral Test with Riemann Sums

    Homework Statement I've seen two methods that prove the integral test for convergence, but I fear they contradict each other. Each method uses an improper integral where the function f(x) is positive, decreasing, and continuous and f(x) = an. What confuses me is one method starts off the...
  25. H

    Definite Integral of the Natural Log of a Quadratic using Riemann Sums

    Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...
  26. S

    Definite integral using Riemann sums?

    I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for...
  27. A

    Question related to Riemann sums, sups, and infs of bounded functions

    Can someone give me an example of a bounded function f defined on a closed interval [a,b] such that f does not attain its sup (or inf) on this interval? Obviously, f cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous...
  28. 1

    Riemann Sums and Integrals, feel lost without actual functions

    Homework Statement At my old university, Calculus was taught much differently than it is where I am now. My old school focused on numerical things, which this school focuses much more on pictures, abstract, etc. and it's very difficult for me. At my old school, we were given a shape...
  29. C

    Infinite riemann sums discrepancy

    Hello. I have to solve some integrals using both the standard theorem of calculus and infinite Riemann sums. \int_{1}^{7} (x^2-4x+2) dx = \lim_{n \to \infty } \sum f(x_i)\Delta x_i = \lim_{n \to \infty } \sum (x_i^2 - 4x_i + 2)6/n Evaluating the definite integral results in an answer of 30...
  30. K

    Riemann Sums: Finding the Limit as n Approaches Infinity

    Homework Statement Identify an=the summation from k=1 to n of (2n)/(4k2+1) as a Riemann sum of an appropriate function on an appropriate interval and find the limit as n approaches infinity of an. Homework Equations There is no interval givien so I assume its from 0 to 1. The...
  31. M

    Approximating integral using riemann sums

    Homework Statement f: [0,1] -> Reals, f(x) = 3-x2 P={0,1/2,1} Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error. Homework Equations The Attempt at a Solution So I tried finding the upper Riemann...
  32. M

    Absolute value of riemann sums

    Homework Statement I'm trying to prove that Sp|f| - sp|f| \leq Spf - spf Where P is a partition of [a,b] and f is function that is riemann integrable. Homework Equations The Attempt at a Solution So I get to a point where M = supf(x) and m = inff(x) then |M|(b-a) - |m|(b-a)...
  33. A

    Left endpoint approximation & Riemann Sums (Sigma)

    1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...
  34. A

    Why Use Supremum Instead of Maximum in Riemann Sums?

    Let f be a Riemann integrable function defined on an interval [a,b], and let P = \{a = x_0 < x_1 < \ldots < x_n = b\} be a partition of [a,b]. I don't understand why the definition of (say) the upper Riemann sum of f associated with P is always given as U(f,P) = \sum_{i=1}^n M_i (x_i -...
  35. W

    Calc 1 Riemann Sums w/ velocity and distance

    Homework Statement This is somewhat a repost... except I have figured out some of it and I have cleaned up the question. Your task is to estimate how far an object traveled during the time interval 0<= t >= 8 , but you only have the following data about the velocity of the object...
  36. W

    Riemann sums with velocity and distance.

    Homework Statement I really need help starting this problem as I am not sure what to do. Your task is to estimate how far an object traveled during the time interval 0t8 , but you only have the following data about the velocity of the object. time (sec) 0 1 2 3 4 5 6 7 8...
  37. P

    Can Riemann Sums Accurately Account for Yearly Wind Turbine Hours?

    Homework Statement The following table shows the power produced by a 600kW wind turbine at the given wind speed and the number of hours the wind blows at that speed. a) Plot the power characteristic as a function of wind speed. b) Plot the wind duration curve as a function of wind speed...
  38. C

    (revised+re-post)Upper and Lower sums & Riemann sums

    http://img156.imageshack.us/i/17818455.jpg/ http://img215.imageshack.us/i/53355598.jpg/ http://img509.imageshack.us/i/11493310.jpg/ If you look at the above, I have underlined the problem that I am having. So, my first question is, where are these inequalities coming from? If you do have...
  39. C

    Upper and Lower sums & Riemann sums

    Homework Statement Homework Equations The Attempt at a Solution I have attachments that can answer the above template, and please look at the attachments if you are trying to help me. I have two questions regarding upper and lower sums & Riemann sums. So, the attachments 1 & 2 are...
  40. J

    Mathematica Help with Right and Left Riemann Sums

    Homework Statement I am given a left riemann sum program module in Mathematica and need to convert it into the right riemann sum. The program takes values for x and f/x and the partition and graphs on a certain interval provided. leftRiemannGraph[f_, a_, b_, n_] := Module[{expr}, expr[1]...
  41. K

    Need some help with Riemann Sums.

    Need some urgent help with Riemann Sums. Homework Statement PART A: In all of this question, let I = \int ^{2}_{-2} f(x)dx where f(x) = -2x + 1 Evaluate I. PART B: Use the defintion of the definite integral to evaluate I. i.e Riemann Sum. Homework Equations The...
  42. U

    Solving Riemann Sums for \int_0^{2\pi} x^{2}sin(x)\,dx | Homework Help

    Homework Statement Express the integral as a limit of Riemann sums. Do not evaluate the limit. Homework Equations \int_0^{2\pi} x^{2}sin(x)\,dx The Attempt at a Solution I really don't know where to start...any help getting me started would be highly appreciated!
  43. S

    Real Analysis: Finding the Limit of a Riemann Sum

    Homework Statement Find the limit, as n -> infinity, of \sum_{k=1}^nk3/n4 Homework Equations Riemann sum: S(f, \pi, \sigma) = \sum_{k=1}^nf(\xi)(xk - xk-1) The Attempt at a Solution My guess is that I should try to put this sum in terms of a Riemann sum, and then taking n -> infinity will...
  44. G

    Why Are Riemann Sums Confusing Without Function Curves?

    Alright, I started doing Riemann sums and I am ripping my hair out in frustration. I just can't wrap my head around how I'm supposed to do it, and my woefully vague textbook isn't helping either. I'm wondering how I'm supposed to solve a Riemann sum question with sigma notation (no limits), and...
  45. A

    Integral Question (Riemann Sums)

    This is from a final exam on the MIT Open Course Ware site for Single Variable Calculus Homework Statement (a)(5 points) Write down the general formula for the Riemann sum approximating the Riemann integral, 1 \int f(x)dx 0 for the partition of [0,1] into n subintervals of equal...
  46. B

    Please HELP Don't Understand Simple Concept on Riemann Sums

    Please HELP...Don't Understand Simple Concept on Riemann Sums Can someone please explain this to me... The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. That is, ||Triangle|| approaches 0 implies that n approaches infinity. I thought...
  47. W

    Solving the Definite Integral Using Right Riemann Sums

    Homework Statement The following sum \sqrt{9 - \left(\frac{3}{n}\right)^2} \cdot \frac{3}{n} + \sqrt{9 - \left(\frac{6}{n}\right)^2} \cdot \frac{3}{n} + \ldots + \sqrt{9 - \left(\frac{3 n}{n}\right)^2} \cdot \frac{3}{n} is a right Riemann sum for the definite integral. Solve as n->infinity...
  48. G

    Riemann Sums infinite strips

    Use the riemann sums model to estimate the area under the curve f(x) = x^2, between x =2 and x = 10, using an infinite number of strips. Be sure to include appropriate diagriams and full explanation of the method of obtaining all numerical values, full working and justification. Does anybody...
  49. Apost8

    Integration or Riemann Sums: Which is More Effective for Numerical Integration?

    Is there ever a situation where it is more appropriate/advantageous to use Riemann summation as opposed to evaluating an integral, or is Riemann summation merely taught in order to help the student to understand what's going on?
  50. T

    Proving Log 2 with Riemann Sums

    I have been working on this problem for a while. I am supposed to prove that log 2 = \lim_{n \rightarrow \infty} \frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{2^n}. The problem is that I have a hard time figuring out how I am supposed to prove that something is equal to a transcendental...