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  2. Riemann sum - Wikipedia

    en.wikipedia.org/wiki/Riemann_sum

    While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.

  3. Fundamental theorem of calculus - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of...

    A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function. We are describing the area of a rectangle, with the width times the height, and we are adding the areas together.

  4. File:Riemann sum convergence.svg - Wikipedia

    en.wikipedia.org/wiki/File:Riemann_sum...

    An example of Riemann sums for the integral ((+ (+ (+))) +), sampling each interval at right (blue), minimum (red), maximum (green), or left (yellow). Convergence of all four choices to 3.76 occurs as number of intervals increases from 2 to 10 (and implicitly, to ∞).

  5. Numberphile - Wikipedia

    en.wikipedia.org/wiki/Numberphile

    Numberphile is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. [2] [3] In the early days of the channel, each video focused on a specific number, but the channel has since expanded its scope, [4] featuring videos on more advanced mathematical concepts such as Fermat's Last Theorem, the Riemann hypothesis [5] and Kruskal's tree ...

  6. Riemann integral - Wikipedia

    en.wikipedia.org/wiki/Riemann_integral

    One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.

  7. Bernhard Riemann - Wikipedia

    en.wikipedia.org/wiki/Bernhard_Riemann

    Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma : if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large n .

  8. Poisson summation formula - Wikipedia

    en.wikipedia.org/wiki/Poisson_summation_formula

    For example, can be the real points of and can be the integral points of . In this setting, G {\displaystyle G} plays the role of the real number line in the classical version of Poisson summation, and Γ {\displaystyle \Gamma } plays the role of the integers n {\displaystyle n} that appear in the sum.

  9. Shell integration - Wikipedia

    en.wikipedia.org/wiki/Shell_integration

    The Riemann sum can be thought up as a sum of a number n of rectangles with ever shrinking bases, we might focus on one of them: f ( a + k Δ x ) Δ x {\displaystyle f(a+k\Delta x)\Delta x}