Name: riemann sums and area estimates
Uploaded: 2026-01-12T12:24:23.000Z
Duration: 1 min 29 s
Channel: JoVE

Considérons une fonction sur un intervalle fermé. Pour approximer l’aire sous la courbe, divisez l’intervalle en sous-intervalles égaux, formant des rectangles de largeur égale.Sélectionnez un point arbitraire dans le sous-intervalle et utilisez la valeur de la fonction là comme hauteur du rectangle. En multipliant la hauteur par la largeur, on obtient la surface de ce rectangle. La surface totale sous la courbe est approximée en sommant les aires de tous les rectangles et on parle de somme de Riemann.L’intégrale définie d’une fonction sur un intervalle donné est définie comme la limite de sa somme de Riemann lorsque le nombre de sous-intervalles tend vers l’infini et que la largeur de chaque rectangle devient infinitésimalement petite, à condition que cette limite existe.Par exemple, considérons une voiture voyageant à une vitesse variable. Divisez le temps total de trajet en intervalles de temps égaux.Dans chaque intervalle, sélectionnez un point temporel arbitraire et prenez la vitesse correspondante. Multipliez la vitesse par la largeur de l’intervalle pour estimer la distance parcourue.La somme de ces éléments forme une somme de Riemann, et dans cet exemple, l’aire sous la courbe est physiquement interprétée comme la distance parcourue par la voiture. Lorsque la vitesse reste positive, la distance totale est déterminée en intégrant la fonction de vitesse dans les limites.

intégrale définie

Definite Integral

Consid&#xE9;rez une fonction &#xE0; valeurs r&#xE9;elles d&#xE9;finie sur un intervalle ferm&#xE9;. L&#x2019;un des objectifs fondamentaux du calcul int&#xE9;gral est de d&#xE9;terminer l&#x2019;aire sous le graphe d&#x2019;une telle fonction. Lorsqu&#x2019;un calcul exact n&#x2019;est pas imm&#xE9;diatement possible, cette aire peut &#xEA;tre estim&#xE9;e en divisant l&#x2019;intervalle en un nombre fini de sous-intervalles de m&#xEA;me longueur. Chaque sous-intervalle correspond &#xE0; un rectangle dont la largeur est &#xE9;gale &#xE0; celle du sous-intervalle et dont la hauteur est d&#xE9;termin&#xE9;e par la valeur de la fonction en un point choisi &#xE0; l&#x2019;int&#xE9;rieur de ce sous-intervalle.
Sommes de Riemann et approximation de l&#x2019;aire
Pour chaque sous-intervalle, choisissez un point, et la valeur de la fonction en ce point d&#xE9;termine la hauteur du rectangle. L&#x2019;aire de chaque rectangle est donn&#xE9;e par le produit de sa hauteur et de sa largeur. La somme des aires de tous les rectangles donne une approximation de l&#x2019;aire totale sous la courbe. Cette approximation est appel&#xE9;e une somme de Riemann. &#xC0; mesure que le nombre de sous-intervalles augmente et que leur largeur diminue, l&#x2019;approximation devient en g&#xE9;n&#xE9;ral plus pr&#xE9;cise et refl&#xE8;te plus fid&#xE8;lement la forme de la courbe.
Int&#xE9;grale d&#xE9;finie comme processus de limite
L&#x2019;int&#xE9;grale d&#xE9;finie pr&#xE9;cise l&#x2019;id&#xE9;e de somme de Riemann en examinant ce qui se passe lorsque la partition de l&#x2019;intervalle devient infiniment fine. L&#x2019;int&#xE9;grale d&#xE9;finie d&#x2019;une fonction sur un intervalle donn&#xE9; est la valeur limite des sommes de Riemann lorsque le nombre de rectangles augmente sans borne et que la largeur de chacun devient n&#xE9;gligeable, &#xE0; condition que cette valeur limite existe. Ainsi, l&#x2019;int&#xE9;grale d&#xE9;finie repr&#xE9;sente l&#x2019;aire exacte sous la courbe sur l&#x2019;intervalle consid&#xE9;r&#xE9;.
Interpr&#xE9;tation dans des contextes physiques
Une illustration pratique de ce concept se trouve dans l&#x2019;&#xE9;tude du mouvement. Consid&#xE9;rez une voiture se d&#xE9;pla&#xE7;ant sur une trajectoire rectiligne avec une vitesse qui varie au cours du temps. En divisant le temps total de d&#xE9;placement en segments &#xE9;gaux et en prenant la vitesse &#xE0; un instant choisi &#xE0; l&#x2019;int&#xE9;rieur de chaque segment, vous pouvez estimer la distance parcourue. La somme de ces distances donne une somme de Riemann qui approxime la distance totale parcourue. Lorsque la vitesse reste positive sur tout l&#x2019;intervalle, l&#x2019;int&#xE9;grale d&#xE9;finie de la fonction vitesse correspond &#xE0; la distance totale parcourue par la voiture.

Intégrale définie

Consid&#xE9;rez une fonction &#xE0; valeurs r&#xE9;elles d&#xE9;finie sur un intervalle ferm&#xE9;. L&#x2019;un des objectifs fondamentaux du calcul ...

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riemann sums and area estimates

Riemann sums and area estimates are a key part of calculus for finding the area under a curve. When a real-valued function is defined on a closed interval, the area under its graph can be approximated if an exact calculation is not easy to do.

The interval is split into equal subintervals. Each subinterval becomes a rectangle. The rectangle’s width is the length of the subinterval, and its height comes from the function value at one chosen point inside that part of the interval. Multiplying height by width gives the area of each rectangle.

Adding the areas of all the rectangles gives a Riemann sum. This sum is an approximation of the total area under the curve. As the number of subintervals increases and the widths get smaller, the estimate usually becomes more accurate because it follows the shape of the curve more closely.

The definite integral builds on this idea by taking the limit of the Riemann sums. It looks at what happens as the partition of the interval becomes infinitely fine and the rectangle widths become negligibly small, provided that the limit exists. In that case, the definite integral gives the exact area under the curve over the interval.

This idea also has a physical meaning in motion. A car moving along a straight path with changing velocity can be studied by dividing the travel time into equal parts. Using the velocity at a chosen time in each part gives an estimate for the distance traveled in that segment. Adding those distances gives a Riemann sum for total distance, and when the velocity stays positive, the definite integral of velocity over time matches the total distance covered.

Riemann Sums and Area Estimates

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key terms and concepts

Learn how Riemann sums use rectangles to estimate area under a curve and how the definite integral gives the exact value as the intervals shrink.

Riemann Sums and Area Estimates | 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

Riemann Sums and Area Estimates

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