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. stevmg

    Riemann sum and anti-derivative

    How do you mathematically equate a Riemann sum as area under the curve to an anti-derivative? How do you prove that, theoreticlly, the one is equalent to the other? Assuming the function is continuous between points a and b, there is always a Riemann sum and thus the function is integrable...
  2. J

    No use of the tangent in Riemann sum?

    Where is the use of the "tangents at every point on the curve" in the Riemann sum? Riemann sum allows us to find the area of under the curve, and this involves only the height of each rectangle (i.e. the function f(x) at each x), and the width (i.e. the x), and the two are multiplied together...
  3. R

    Surface Area of Sphere as a Riemann Sum

    Homework Statement How do you solve the surface area of a sphere using Riemann Sums?Homework Equations The Attempt at a Solution I started out with 2 * (lim n->∞ [ (i=1 to n) ∑ [ 2*pi*(√(r^2 - (i/rn)^2))*(r/n) ] ]) where the summation is the surface area of the cylinders (or discs) inside a...
  4. F

    How Do You Calculate Riemann Sums for sqrt(x) with a Squared Partition?

    Hello everyone, I have passed my integral calculus class and it's been a little while so I don't really remember everything. Can anyone help me out with this? Homework Statement Let f(x) = sqrt(x), x E [0,1] and P=\left \{ 0,\left ( \frac{1}{n} \right )^{2}, \left ( \frac{2}{n} \right...
  5. T

    Fortran Find Area Under Curve w/ Fortran Riemann Sums

    Hello, for my FORTRAN class it wants me to use the method of Riemann's sums to find the area under the curve for the function f(x) = -(x-3)**2 +9, and stop when successive iterations yield a change of less than 0.0000001. I know I am going to have to used double precision. I am just confused...
  6. B

    Riemann Sum: Solve for Area Under Curve 0 to 18

    Riemann sum help! Homework Statement Use Riemann sum with ci= i3/n3 f(x)= \sqrt[3]{x} +12 from x=0 to x=18 n= 6 subintervals Approximate the sum using Riemann's Sum Homework Equations \Sigma f(ci) \Delta xi is the equation for riemanns sum i think The Attempt at a Solution i...
  7. G

    How to Use Riemann Sums and Integrals to Estimate and Evaluate Functions

    Homework Statement (x, f(x)) (2,1) (3,4) (5,-2) (8,3) (13,6) A) Estimate f '(4). Show work. B) Evaluate the Intergral from 2 to 13 of (3 - 5f '(x))dx. show work C) Use left riemann sum with subintervals indicated bye the data in the table to apporoximate the intergral from 2 to 13 of...
  8. D

    Evaluating Riemann Sum f(x,y) - 4x^2+y

    The problem says: evaluate 4x^2+y by breaking into four congruent subrectangles and evaluating at the midpoints, 1=<x<=5 0=<y<=2 When i setup the rectangles these are my coordinates: (1,1/2),(1,3/2),(3,1/2),(3,3/2) and delta A = 2 My answer comes out to be 168...
  9. M

    Calculating the Area of a Strip Using a Riemann Sum

    Homework Statement Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown in the figure below where the upper line is defined by 6x + y = 12 and the other line is defined by y=x^2-4. The figure, which I can't get on here, is just the...
  10. Y

    Riemann sum question, with picture

    Homework Statement http://img4.imageshack.us/img4/898/integerqj5.jpg Homework Equations The Attempt at a Solution It does appear to be a Riemann sum, I figured the 1/n is probably the width of the intervals and the sum in brackets is related to the sums of the heights of the rectangles. But my...
  11. N

    Comp Sci Calculating pi using Riemann Sum and Fortran77

    My assignment: Solve for pi using a Riemann Sum with n= 40,000,000. The function is the antiderivate of 4/(1+x^2) dx. The bounds are from 0 to 1. Solving this gives you pi. Anyone know how to do this? Preferably with fortran77?
  12. R

    Writing a Riemann Sum w/out Sumation signs

    Homework Statement In this problem you will calculate ∫0,4 ( [(x^2)/4] − 7) dx by using the definition ∫a,b f(x) dx = lim (n → ∞) [(n ∑ i=1) (f(xi) ∆x)] The summation inside the brackets is Rn which is the Riemann sum where the sample points are chosen to be the right-hand endpoints...
  13. T

    Fortran What is the first approximation for Riemann Sum using Fortran 90?

    [SOLVED] Riemann Sum with Fortran 90 My assignment: Use Reimann Sums to estimate pi to 6 decimal places (ie: you can stop when successive iterations yield a change of less than 0.000001. For the Reimann Sums solution, an iteration equals 2X the number of segments as the trial before. Print out...
  14. P

    Evaluating a Riemann Sum for $\int^{-2}_{5} t^2 + 6t - 4 dt$

    [SOLVED] Riemann sum Important stuff: \sum i^2 = \frac{n(n+1)(2n+1)}{6} \sum i = \frac{n(n+1)}{2} And the solution: (Where I write "lim" I mean limit as n-->infinity. Where I write the summation sign I mean from i=1 to n.) lim \sum t^2 + 6t - 4 \Delta t \Delta t = \frac{5 -...
  15. S

    Find the Riemann sum for this integral

    Find the Riemann sum for this integral using the right-hand sums for n=4 Find the Riemann sum for this same integral, using the left-hand sums for n=4 Sorry the integral is attatched. I don't know how to get it onto here.
  16. T

    Finding Riemann Sum for f(x)=3x^2+3

    Homework Statement Find the Riemann sum associated with f(x)=3 x^2 +3 ,\quad n=3 and the partition x_0=0,\quad x_1=3,\quad x_2=4,\quad x_3=6,\qquad \mbox{ of } [0,6] (a) when x_k^{*} is the right end-point of [x_{k-1},x_k]. . (b) when x_k^{*} is the mid-point of [x_{k-1},x_k]...
  17. 4

    Convergence and Divergence of Riemann Sum Series with Varying p

    For what values of p>0 does the series Riemann Sum [n=1 to infinity] 1/ [n(ln n) (ln(ln n))^p] converge and for what values does it diverge? How do i do this question? Would somebody please kindly show me the steps? Do i use the intergral test?
  18. C

    Riemann Sum Question: Is my Answer Correct?

    Hello, just going through some Riemann sum problems before I hit integrals and I am like 99% sure that this answer from my text is wrong but I want to make sure. It's not really an important question so if you have better things to do, help the next guy :) But checking this over would be...
  19. W

    What is the definition of S in the proof for convergence of Riemann sum?

    In a book of mine, the author proceeds to the proof that a Riemann sum in a interval [a,b] must converge by proving that for S_m and S_n (m>n) where the span of the subdivisions is suffiencienly small, then |S_m - S_n)| < e(b-a) Where e can assume infinitly small values in dependence of...
  20. quasar987

    Riemann sum of derivative (something like that)

    Hi, maybe someone can help. When I think about it, I'm pretty sure that the following is true: Let c be a curve parametrized by t\in [a,b], let \sigma = \{t_0,...,t_N\} be a partition of [a,b] and \delta_{\sigma}=\max_{0\leq k \leq N-1}(t_{k+1}-t_k). Also define \Delta t_k=t_{k+1}-t_k Then...
  21. S

    Evaluating Exact Area using Riemann Sum and Limit on y = 2x^2 + 4 from 0 to 1

    Use the Riemann sum and a limit to evaluate the exact area under the graph of y = 2x^2 + 4 on [0, 1] I know how to do this normally but now they ask to do it w/ a limit and I'm not sure how. (LaTex corrected by HallsofIvy.)
  22. P

    Calculating the Riemann Sum of (cos1)^x

    hi, is it possible to find the riemann sum of (cos1)^x? it looks divergent to me can someome please help me... even if it is convergent, i don't know how to find the sum of a trigonometric function
  23. X

    Proof w/ natural log and Riemann Sum

    Problem states: (A) Use mathematical induction to prove that for x\geq0 and any positive integer n. e^x\geq1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!} (B) Use part (A) to show that e>2.7. (C) Use part (A) to show that \lim_{x\rightarrow\infty} \frac{e^x}{x^k} =...
  24. N

    Riemann sum / difference quotient

    How does the difference quotient undo what the Riemann sum does or vice versa. In terms of the two formulas? I would assume that working a difference quotient backwards would be similar to working a Riemann sum forward, but in reality as the operations go this couldn't be further from the...
  25. G

    How to Approximate the Area of a Semi-Circle Using Riemann Sums

    Can anyone please direct me in the right way on working out the approximate area of a semi-circle with equation y = (r^2 - x^2)^0.5, by using a Riemann Sum
  26. D

    Can anyone help with finding the Riemann sum for f(x)=x^3?

    Hello there, can anyone help me here as I'm finding it difficult to tackle this question. Consider f(x)=x^3 on the interval [1,5]. Find the Riemann sum for the equipartition P=(1,2,3,4,5) into 4 intervals with x_i^* being the right-hand endpoints (ie. x_i=a+hi) Then find a formula for the...
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