Name: finding water volume with integration
Uploaded: 2026-01-12T12:24:27.000Z
Duration: 1 min 25 s
Channel: JoVE

L’eau s’écoule dans un grand réservoir industriel, mais le débit d’entrée n’est pas constant. Au départ, la pompe fournit de l’eau à 5m 3/s.L’apport d’eau augmente à un rythme constant de deux mètres cubes par seconde grâce aux ajustements continus du système.Ce changement constant fait du débit d’eau une fonction linéaire du temps, où 2 est la pente indiquant le taux d’augmentation, et 5 est le débit d’entrée initial.L’objectif est de déterminer le volume total d’eau stocké dans le réservoir à un moment donné. Cela nécessite une intégration indéfinie, qui détermine le volume total d’eau en intégrant le débit d’eau au fil du temps.Au lieu d’analyser l’afflux d’entrée à des instants précis, l’intégration fournit une fonction unique pour le volume total d’eau par rapport au temps.L’intégration du débit d’entrée linéaire introduit un terme quadratique, qui montre la croissance accélérée du volume d’eau, tandis que le terme linéaire montre la croissance constante à partir du débit d’entrée initial.Enfin, la constante d’intégration est donnée par la fonction de volume à t égal à zéro, ce qui correspond au volume initial d’eau présent dans le réservoir.

intégrales indéfinies

Le d&#xE9;bit d&#x2019;eau entrant dans une cuve de stockage n&#x2019;est pas constant, mais augmente au fil du temps. Au d&#xE9;part, la pompe d&#xE9;bite de l&#x2019;eau &#xE0; un d&#xE9;bit de 5 L/min. Toutefois, ce d&#xE9;bit augmente de 2 L/min &#xE0; chaque minute suppl&#xE9;mentaire, en raison d&#x2019;une hausse de la pression ou d&#x2019;ajustements du syst&#xE8;me. Cette situation peut &#xEA;tre d&#xE9;crite math&#xE9;matiquement par une fonction lin&#xE9;aire :\begin{equation}f(t) = 2t + 5\end{equation}Il est n&#xE9;cessaire d&#x2019;int&#xE9;grer la fonction d&#xE9;crivant le d&#xE9;bit afin de d&#xE9;terminer le volume total d&#x2019;eau ajout&#xE9; &#xE0; la cuve au cours du temps. Le volume total d&#x2019;eau V(t) est obtenu en calculant l&#x2019;int&#xE9;grale suivante :\begin{equation}V(t) = \int (2t + 5) \, dt\end{equation}Lorsque la r&#xE8;gle des puissances est appliqu&#xE9;e pour &#xE9;valuer l&#x2019;int&#xE9;grale, des termes quadratiques et lin&#xE9;aires apparaissent, traduisant l&#x2019;&#xE9;volution du d&#xE9;bit :\begin{equation}V(t) = t^2 + 5t + C\end{equation}Le terme t^2 appara&#xEE;t en raison de l&#x2019;augmentation progressive du d&#xE9;bit, montrant que l&#x2019;accumulation d&#x2019;eau suit une loi quadratique au cours du temps. Le terme 5t repr&#xE9;sente la composante de d&#xE9;bit constant pr&#xE9;sente d&#xE8;s le d&#xE9;but. La constante C correspond au volume initial d&#x2019;eau contenu dans la cuve avant le d&#xE9;but du processus de remplissage. Dans la mesure o&#xF9; l&#x2019;int&#xE9;gration permet de d&#xE9;terminer le volume total en sommant le d&#xE9;bit instantan&#xE9;, cette &#xE9;quation fournit une fonction compl&#xE8;te d&#xE9;crivant la quantit&#xE9; totale d&#x2019;eau pr&#xE9;sente dans la cuve &#xE0; tout instant.

Intégrales indéfinies

Le d&#xE9;bit d&#x2019;eau entrant dans une cuve de stockage n&#x2019;est pas constant, mais augmente au fil du temps. Au d&#xE9;part, la pompe ...

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Calculus

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Integrals

Intégrales

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Indefinite Integrals

finding water volume with integration

Integration can be used to find the total volume of water in a tank when the inflow rate changes over time. In this example, the pump starts at 5 L/min. The inflow rate then increases by 2 L/min for each additional minute. That changing rate can be written as a linear function.

To measure how much water has been added, the inflow rate function is integrated over time. The total water volume V(t) comes from this integral. When the power rule is used, the result includes both a quadratic term and a linear term. The t^2 term comes from the increasing inflow rate, so the water amount grows in a quadratic pattern. The 5t term comes from the steady inflow that is present from the start.

The constant C represents the initial volume of water already in the tank before the inflow begins. In this way, integration adds up the instantaneous rate of change to give a function for the total amount of water at any time.

Finding Water Volume with Integration

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key terms and concepts

See how integration models water flowing into a tank when the inflow rate changes over time, using a linear rate, the power rule, and a constant.

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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

Finding Water Volume with Integration

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