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

A água flui para um grande tanque industrial de armazenamento, mas a taxa de entrada não é constante. Inicialmente, a bomba fornece água a 5m 3/s.O fluxo de água está aumentando a uma taxa constante de dois metros cúbicos por segundo devido a ajustes contínuos do sistema.Essa variação constante faz com que a taxa de entrada de água seja uma função linear pelo tempo, onde 2 é a inclinação que mostra a taxa de aumento, e 5 é a taxa inicial de entrada.O objetivo é encontrar o volume total de água armazenada no tanque em qualquer momento. Isso requer integração indefinida, que determina o volume total de água integrando a taxa de entrada ao longo do tempo.Em vez de analisar o fluxo de entrada em instantes específicos, a integração fornece uma única função para o volume total de água em relação ao tempo.A integração da taxa de entrada linear introduz um termo quadrático, que mostra o crescimento acelerado do volume de água, enquanto o termo linear mostra o crescimento constante a partir da taxa inicial de entrada.Por fim, a constante de integração é dada pela função de volume em t igual a zero, que corresponde ao volume inicial de água presente no tanque.

integrais indefinidas

A vaz&#xE3;o de entrada de &#xE1;gua em um tanque de armazenamento n&#xE3;o &#xE9; constante, mas aumenta com o tempo. Inicialmente, a bomba fornece &#xE1;gua a uma vaz&#xE3;o de 5 L/min. No entanto, a vaz&#xE3;o aumenta em 2 L/min a cada minuto adicional devido ao aumento da press&#xE3;o ou a ajustes no sistema. Este cen&#xE1;rio pode ser descrito matematicamente por uma fun&#xE7;&#xE3;o linear:\begin{equation}f(t) = 2t + 5\end{equation}&#xC9; necess&#xE1;rio integrar a fun&#xE7;&#xE3;o da vaz&#xE3;o para medir o volume total de &#xE1;gua adicionado ao tanque ao longo do tempo. O volume total de &#xE1;gua V(t) &#xE9; obtido realizando a seguinte integral:\begin{equation}V(t) = \int (2t + 5) \, dt\end{equation}Quando a regra da pot&#xEA;ncia &#xE9; aplicada para avaliar a integral, surgem termos quadr&#xE1;ticos e lineares, refletindo a varia&#xE7;&#xE3;o da taxa de entrada:\begin{equation}V(t) = t^2 + 5t + C\end{equation}O termo t^2 surge devido ao aumento da vaz&#xE3;o, demonstrando que o ac&#xFA;mulo de &#xE1;gua segue um padr&#xE3;o quadr&#xE1;tico ao longo do tempo. O termo 5t representa o componente de fluxo constante que existe desde o in&#xED;cio. A constante C representa o volume inicial de &#xE1;gua no tanque antes do in&#xED;cio do processo de entrada. Como a integra&#xE7;&#xE3;o determina o volume total ao somar a taxa instant&#xE2;nea de varia&#xE7;&#xE3;o, esta equa&#xE7;&#xE3;o fornece uma fun&#xE7;&#xE3;o completa que descreve a quantidade total de &#xE1;gua no tanque em qualquer instante.

Integrais Indefinidas

A vaz&#xE3;o de entrada de &#xE1;gua em um tanque de armazenamento n&#xE3;o &#xE9; constante, mas aumenta com o tempo. Inicialmente, a bomba fornece ...

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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.

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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