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

El agua fluye hacia un gran depósito industrial de almacenamiento, pero el caudal no es constante. Inicialmente, la bomba suministra agua a 5m 3/s.El caudal de agua aumenta a una tasa constante de dos metros cúbicos por segundo debido a los ajustes continuos del sistema.Este cambio constante hace que la tasa de entrada de agua sea una función lineal del tiempo, donde 2 es la pendiente que indica la tasa de aumento, y 5 es la tasa inicial de entrada.El objetivo es encontrar el volumen total de agua almacenada en el tanque en cualquier momento. Esto requiere una integración indefinida, que determina el volumen total de agua integrando la tasa de entrada de agua a lo largo del tiempo.En lugar de analizar el flujo de entrada en instantes específicos, la integración proporciona una función única para el volumen total de agua en relación con el tiempo.Integrar la tasa de entrada lineal introduce un término cuadrático, que muestra el crecimiento acelerado del volumen de agua, mientras que el término lineal muestra el crecimiento constante desde la tasa inicial de entrada.Finalmente, la constante de integración está dada por la función de volumen en t igual a cero, que corresponde al volumen inicial de agua presente en el tanque.

integrales indefinidas

El caudal de entrada de agua en el dep&#xF3;sito de almacenamiento no es constante, sino que aumenta con el tiempo. Inicialmente, la bomba impulsa el agua a un caudal de 5 L/min. Sin embargo, el caudal aumenta en 2 L/min por cada minuto adicional debido al incremento de presi&#xF3;n o a los ajustes del sistema. Este escenario se puede describir matem&#xE1;ticamente mediante una funci&#xF3;n lineal:\begin{equation}f(t) = 2t + 5\end{equation}Es necesario integrar la funci&#xF3;n del caudal de entrada para determinar el volumen total de agua a&#xF1;adido al dep&#xF3;sito a lo largo del tiempo. El volumen total de agua V(t) se obtiene mediante la siguiente integral:\begin{equation}V(t) = \int (2t + 5) \, dt\end{equation}Al aplicar la regla de potencia para evaluar la integral, surgen t&#xE9;rminos cuadr&#xE1;ticos y lineales que reflejan la variaci&#xF3;n del caudal de entrada:\begin{equation}V(t) = t^2 + 5t + C\end{equation}El t&#xE9;rmino t^2 surge debido al incremento del caudal de entrada, lo que indica que la acumulaci&#xF3;n de agua sigue un patr&#xF3;n cuadr&#xE1;tico a lo largo del tiempo. El t&#xE9;rmino 5t representa el componente de caudal constante que existe desde el principio. La constante C representa el volumen inicial de agua en el dep&#xF3;sito antes de que comience el proceso de entrada. Dado que la integraci&#xF3;n determina el volumen total sumando el caudal instant&#xE1;neo, esta ecuaci&#xF3;n proporciona una funci&#xF3;n que describe completamente la cantidad total de agua en el dep&#xF3;sito en un momento dado.

Integrales Indefinidas

El caudal de entrada de agua en el dep&#xF3;sito de almacenamiento no es constante, sino que aumenta con el tiempo. Inicialmente, la bomba impulsa el agua ...

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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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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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