Name: definite integrals with antiderivatives
Uploaded: 2026-01-12T12:24:04.000Z
Duration: 1 min 29 s
Channel: JoVE

La seconde partie du théorème fondamental du calcul relie les antidérivées aux intégrales définies. Il trouve l’accumulation totale exacte du taux d’une fonction sur un intervalle spécifique.Si une fonction est continue sur un intervalle fermé, l’intégrale définie est égale à la valeur de l’antidérivée à la limite supérieure moins sa valeur à la limite inférieure.Cela évite les approximations de somme de Riemann, qui estiment l’aire en utilisant de nombreux rectangles étroits sous la courbe.L’antidérivée choisie n’affecte pas le résultat final. Par exemple, G de x peut différer d’une autre antidérivée par une constante C, mais C s’annule lorsqu’on soustrait des valeurs aux extrémités de l’intervalle.Une application courante consiste à trouver le déplacement net d’un objet en utilisant sa fonction de vitesse.Le déplacement net pour un intervalle de temps donné, allant de 1 seconde à 4 secondes, est l’aire sous le graphique vitesse-temps.Pour calculer cela, on trouve d’abord l’antidérivée de la fonction de vitesse — cela donne la fonction position. Ensuite, en utilisant le théorème, évaluez la position aux extrémités de l’intervalle. Soustrayons la valeur de la fonction position à 1 seconde de sa valeur à 4 secondes pour obtenir le déplacement net.

théorème fondamental de l’analyse ii

En analyse, le calcul de l&#x2019;aire sous une courbe continue a &#xE9;t&#xE9; profond&#xE9;ment simplifi&#xE9; gr&#xE2;ce &#xE0; l&#x2019;application du th&#xE9;or&#xE8;me fondamental de l&#x2019;analyse, partie 2. Plut&#xF4;t que de s&#x2019;appuyer sur le proc&#xE9;d&#xE9; de passage &#xE0; la limite consistant &#xE0; sommer une infinit&#xE9; de rectangles infinit&#xE9;simaux, ce th&#xE9;or&#xE8;me permet une &#xE9;valuation directe au moyen des primitives, simplifiant ainsi le calcul des int&#xE9;grales d&#xE9;finies.Le th&#xE9;or&#xE8;me fondamental de l&#x2019;analyse, partie 2, &#xE9;nonce que si une fonction f(x) est continue sur un intervalle ferm&#xE9; [a, b], alors l&#x2019;int&#xE9;grale d&#xE9;finie de f sur cet intervalle peut &#xEA;tre calcul&#xE9;e &#xE0; l&#x2019;aide de n&#x2019;importe laquelle de ses primitives F(x). Math&#xE9;matiquement, cela s&#x2019;exprime ainsi :\begin{equation*}\int_a^b f(x)\,dx = F(b) - F(a)\end{equation*}Ce r&#xE9;sultat repose sur le fait que toutes les primitives d&#x2019;une fonction ne diff&#xE8;rent que par une constante. Comme cette constante s&#x2019;&#xE9;limine lors de l&#x2019;&#xE9;valuation de la diff&#xE9;rence F(b) &#x2212; F(a), toute primitive peut &#xEA;tre utilis&#xE9;e pour calculer l&#x2019;int&#xE9;grale, ce qui accro&#xEE;t consid&#xE9;rablement la flexibilit&#xE9; du calcul.Il est essentiel de noter que la fonction f(x) doit &#xEA;tre continue sur tout l&#x2019;intervalle [a, b]. La pr&#xE9;sence de discontinuit&#xE9;s ou de points o&#xF9; la fonction n&#x2019;est pas d&#xE9;finie sur cet intervalle rend le th&#xE9;or&#xE8;me inapplicable. Dans de tels cas, l&#x2019;int&#xE9;grale d&#xE9;finie peut ne pas &#xEA;tre bien d&#xE9;finie avec cette m&#xE9;thode, et il est alors n&#xE9;cessaire de recourir &#xE0; des techniques alternatives ou &#xE0; une int&#xE9;gration par morceaux soigneusement justifi&#xE9;e.Ce th&#xE9;or&#xE8;me transforme fondamentalement l&#x2019;int&#xE9;gration, en la faisant passer d&#x2019;un processus fond&#xE9; sur la sommation de quantit&#xE9;s infinit&#xE9;simales &#xE0; l&#x2019;&#xE9;valuation de valeurs de la fonction &#xE0; des bornes pr&#xE9;cises. Il constitue une pierre angulaire du calcul int&#xE9;gral et permet d&#x2019;obtenir des solutions analytiques &#xE0; de nombreux probl&#xE8;mes en math&#xE9;matiques, en physique et en sciences de l&#x2019;ing&#xE9;nieur.

Théorème fondamental de l’analyse II

En analyse, le calcul de l&#x2019;aire sous une courbe continue a &#xE9;t&#xE9; profond&#xE9;ment simplifi&#xE9; gr&#xE2;ce &#xE0; l&#x2019;application du ...

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Fundamental Theorem of Calculus II

definite integrals with antiderivatives

The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative.

If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because any two antiderivatives of the same function differ only by a constant.

That constant does not change the final answer. When F(b) - F(a) is calculated, the constant cancels out. This gives flexibility, because any antiderivative can be used to find the integral.

The theorem only applies when the function is continuous on the entire interval [a, b]. If the function has a discontinuity or is undefined somewhere inside the interval, this method does not work. In those cases, the definite integral may need another approach, such as careful piecewise integration.

This theorem turns integration into a process based on endpoint values instead of infinite sums. It is a key idea in integral calculus and helps solve problems in mathematics, physics, and engineering.

Definite Integrals with Antiderivatives

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The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative. If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because...

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key terms and concepts

Learn how the Fundamental Theorem of Calculus links definite integrals and antiderivatives, and why continuity on [a, b] matters.

Definite Integrals with Antiderivatives | 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

Definite Integrals with Antiderivatives

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