Name: gauss sums for riemann rectangles
Uploaded: 2026-01-12T12:24:24.000Z
Duration: 1 min 22 s
Channel: JoVE

En intégrales définies, les formules de somme pour les entiers, carrés et cubes consécutifs estiment l’aire à l’aide de sommes de Riemann. Ils simplifient les calculs sur une séquence ou un motif.Considérons une section d’un grand théâtre, où la première rangée a un siège, et chaque nouvelle rangée ajoute un siège supplémentaire, formant une séquence allant jusqu’à 35 rangées.Le nombre total de sièges est obtenu en utilisant la formule somme des n premiers entiers consécutifs, où n est le nombre de lignes. Après substitution et résolution, le nombre total de sièges peut être facilement calculé.Un concept similaire apparaît dans les supermarchés, où les canettes sont empilées en pyramide carrée pour être exposées. Chaque couche comporte un nombre carré de boîtes — une sur le dessus, quatre sur la deuxième couche, neuf sur la troisième, et ainsi de suite.Pour déterminer le nombre total de cans dans un affichage à cinq couches, on utilise la formule de la somme des carrés, où n est le nombre de couches. En substituant la valeur de n dans la formule et en simplifiant on obtient le nombre total de boîtes utilisées dans la pyramide.Si l’affichage suit un motif d’empilement basé sur des cubes, la formule de la somme des cubes trouve instantanément le nombre total.

intégrales définies : sommes de puissances

En int&#xE9;gration d&#xE9;finie, les sommes de Riemann permettent d&#x2019;approximer l&#x2019;aire situ&#xE9;e sous une courbe en la divisant en sous-intervalles et en additionnant les aires des rectangles correspondants. Lorsque ces approximations suivent des r&#xE9;gularit&#xE9;s num&#xE9;riques pr&#xE9;visibles, telles que des suites arithm&#xE9;tiques ou polynomiales, les formules de sommation constituent un moyen plus efficace et plus pr&#xE9;cis de calculer le r&#xE9;sultat. En particulier, la somme des entiers cons&#xE9;cutifs, des carr&#xE9;s et des cubes joue un r&#xF4;le essentiel dans la simplification de ces calculs, notamment lorsqu&#x2019;il s&#x2019;agit de partitions uniformes ou de donn&#xE9;es structur&#xE9;es g&#xE9;om&#xE9;triquement.M&#xE9;thode de Gauss pour la somme des entiers cons&#xE9;cutifsCarl Friedrich Gauss a &#xE9;labor&#xE9; une m&#xE9;thode ing&#xE9;nieuse pour calculer la somme des n premiers nombres naturels. Il a observ&#xE9; que le fait d&#x2019;apparier les nombres dans l&#x2019;ordre croissant et d&#xE9;croissant conduit &#xE0; des sommes identiques :\begin{equation*} S = 1 + 2 + 3 + \cdots + (n-2) + (n-1) + n&#xA0; \end{equation*}\begin{equation*} S = n + (n-1) + (n-2) + \cdots + 3 + 2 + 1&#xA0; \end{equation*}En additionnant ces deux expressions terme &#xE0; terme :\begin{equation*} 2S = n(n + 1) \end{equation*}Chaque paire donne pour somme (n + 1), et il existe n paires de ce type ; on obtient donc :\begin{equation*} S = \jfrac{n(n + 1)}{2} \end{equation*}Cette d&#xE9;monstration &#xE9;l&#xE9;gante fournit une m&#xE9;thode rapide et g&#xE9;n&#xE9;ralisable pour d&#xE9;terminer la somme de toute suite arithm&#xE9;tique de raison 1.Ces identit&#xE9;s permettent d&#x2019;obtenir des valeurs exactes sans recourir &#xE0; des additions r&#xE9;p&#xE9;t&#xE9;es et sont fondamentales aussi bien dans les mod&#xE8;les discrets que continus rencontr&#xE9;s en analyse (calcul diff&#xE9;rentiel et int&#xE9;gral) ainsi qu&#x2019;en math&#xE9;matiques appliqu&#xE9;es.

Intégrales définies : sommes de puissances

En int&#xE9;gration d&#xE9;finie, les sommes de Riemann permettent d&#x2019;approximer l&#x2019;aire situ&#xE9;e sous une courbe en la divisant en ...

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Sums of Power

gauss sums for riemann rectangles

Riemann sums use rectangles to estimate the area under a curve in definite integration. The interval is split into subintervals, and the rectangle areas are added together. When those rectangle heights follow a regular pattern, sum formulas can make the calculation faster and more exact.

This page focuses on sums that appear in arithmetic and polynomial sequences. The sum of consecutive integers, squares, and cubes is especially useful when the partition is uniform or when the data has a geometric pattern. These identities replace repeated addition with a direct formula.

Gauss’s method gives a quick way to find the sum of the first n natural numbers. He paired the numbers in forward order and reverse order. Each pair added to n + 1, and there were n such pairs. That idea leads to a general formula for an arithmetic sequence with a common difference of 1.

These sum formulas are important in both discrete and continuous models. They provide exact values without step-by-step counting. In calculus and applied mathematics, they help simplify Riemann sum calculations that build from evenly spaced intervals.

Gauss Sums for Riemann Rectangles

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key terms and concepts

Learn how sum formulas and Gauss’s method simplify Riemann sums by using consecutive integers, squares, and cubes in calculus-style area estimates.

Gauss Sums for Riemann Rectangles | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Gauss Sums for Riemann Rectangles

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