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

L'acqua scorre in un grande serbatoio industriale, ma la portata d'ingresso non è costante. Inizialmente, la pompa fornisce acqua a 5m 3/s.L'afflusso d'acqua aumenta a una velocità costante di due metri cubi al secondo grazie ai continui aggiustamenti del sistema.Questa variazione costante rende la portata di afflusso dell'acqua una funzione lineare del tempo, dove 2 è la pendenza che indica la velocità di aumento, e 5 è la portata di afflusso iniziale.L'obiettivo è trovare il volume totale di acqua immagazzinata nel serbatoio in qualsiasi momento. Ciò richiede un'integrazione indefinita, che determina il volume totale dell'acqua integrando la velocità di afflusso nel tempo.Invece di analizzare l'afflusso in istanti specifici, l'integrazione fornisce una funzione unica per il volume totale dell'acqua rispetto al tempo.Integrando la portata di afflusso lineare si introduce un termine quadratico, che mostra la crescita accelerata del volume d'acqua, mentre il termine lineare mostra la crescita costante dalla portata di afflusso iniziale.Infine, la costante di integrazione è data dalla funzione di volume a t uguale a zero, che corrisponde al volume iniziale d'acqua presente nel serbatoio.

integrali indefiniti

La portata in ingresso dell&#x2019;acqua in un serbatoio di accumulo non &#xE8; costante, ma aumenta nel tempo. Inizialmente, la pompa immette acqua con una portata di 5 L/min. Tuttavia, la portata aumenta di 2 L/min per ogni minuto aggiuntivo, a causa dell&#x2019;aumento della pressione o di regolazioni del sistema. Questo scenario pu&#xF2; essere descritto matematicamente con una funzione lineare:\begin{equation}f(t) = 2t + 5\end{equation}&#xC8; necessario integrare la funzione della portata in ingresso per misurare il volume totale d&#x2019;acqua aggiunto al serbatoio nel tempo. Il volume totale d&#x2019;acqua V(t) si ottiene calcolando il seguente integrale:\begin{equation}V(t) = \int (2t + 5) \, dt\end{equation}Applicando la regola della potenza per valutare l&#x2019;integrale, emergono termini sia quadratici sia lineari, che riflettono la variazione della portata nel tempo:\begin{equation}V(t) = t^2 + 5t + C\end{equation}Il termine t^2 deriva dall&#x2019;aumento della portata, mostrando che l&#x2019;accumulo d&#x2019;acqua segue un andamento quadratico nel tempo. Il termine 5t rappresenta la componente di portata costante presente fin dall&#x2019;inizio. La costante C tiene conto del volume iniziale d&#x2019;acqua nel serbatoio prima dell&#x2019;avvio del processo di immissione. Poich&#xE9; l&#x2019;integrazione determina il volume totale sommando la portata istantanea, questa equazione fornisce una funzione completa che descrive la quantit&#xE0; totale d&#x2019;acqua nel serbatoio in qualunque istante.

Integrali indefiniti

La portata in ingresso dell&#x2019;acqua in un serbatoio di accumulo non &#xE8; costante, ma aumenta nel tempo. Inizialmente, la pompa immette acqua con ...

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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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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...

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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.

Finding Water Volume with Integration | 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

Finding Water Volume with Integration

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