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

Wasser fließt in einen großen industriellen Speichertank, aber die Zuflussrate ist nicht konstant. Anfangs liefert die Pumpe Wasser mit 5 m3/s.Der Wasserzufluss nimmt aufgrund kontinuierlicher Systemanpassungen mit einer konstanten Geschwindigkeit von zwei Kubikmetern pro Sekunde zu.Diese stetige Änderung macht die Wasserzuflussrate zu einer linearen Funktion der Zeit, wobei 2 die Steigung ist, die die Zunahmerate anzeigt, und 5 die Anfangszuflussrate.Das Ziel ist es, das gesamte Wasservolumen zu bestimmen, das zu einem bestimmten Zeitpunkt im Tank gespeichert ist. Dies erfordert eine unbestimmte Integration, die das Gesamtvolumen des Wassers durch die Integrierung der Wasserzuflussrate über die Zeit bestimmt.Anstatt den Zufluss zu bestimmten Zeitpunkten zu analysieren, bietet die Integration eine einzige Funktion für das gesamte Wasservolumen in Bezug auf die Zeit.Die Integration der linearen Zuflussrate führt einen quadratischen Term ein, der das beschleunigte Wachstum des Wasservolumens zeigt, während der lineare Term das konstante Wachstum ab der anfänglichen Zuflussrate anzeigt.Schließlich wird die Integrationskonstante durch die Volumenfunktion bei t gleich null gegeben, was dem Anfangsvolumen des im Tank vorhandenen Wasservolumens entspricht.

unbestimmte integrale

Der Wasserzufluss in einen Wassertank ist nicht konstant, sondern nimmt im Laufe der Zeit zu. Anfangs liefert die Pumpe Wasser mit einer F&#xF6;rderrate von 5 L/min. Aufgrund steigenden Drucks im System oder von Systemanpassungen erh&#xF6;ht sich die Zuflussrate jedoch mit jeder weiteren Minute um 2 L/min. Dieses Szenario l&#xE4;sst sich mathematisch durch eine lineare Funktion beschreiben:\begin{equation}f(t) = 2t + 5\end{equation}Um das dem Tank insgesamt zugef&#xFC;hrte Wasservolumen &#xFC;ber die Zeit zu bestimmen, muss die Zuflussfunktion integriert werden. Das Gesamtvolumen V(t) ergibt sich aus folgendem Integral:\begin{equation}V(t) = \int (2t + 5) \, dt\end{equation}Bei Anwendung der Potenzregel zur Auswertung des Integrals treten sowohl quadratische als auch lineare Terme auf, die die &#xC4;nderung der Zuflussrate widerspiegeln:\begin{equation}V(t) = t^2 + 5t + C\end{equation}Der Term t^2 entsteht aufgrund der steigenden Zuflussrate und zeigt, dass die Wasserakkumulation &#xFC;ber die Zeit einem quadratischen Verlauf folgt. Der Term 5t beschreibt die von Anfang an bestehende konstante Zuflusskomponente. Die Konstante C ber&#xFC;cksichtigt das anf&#xE4;ngliche Wasservolumen im Tank vor Beginn des Zuflusses. Da die Integration das Gesamtvolumen durch Summierung der momentanen &#xC4;nderungsraten bestimmt, liefert diese Gleichung eine vollst&#xE4;ndige Funktionsbeschreibung der Gesamtwassermenge im Tank zu jedem beliebigen Zeitpunkt.

Unbestimmte Integrale

Der Wasserzufluss in einen Wassertank ist nicht konstant, sondern nimmt im Laufe der Zeit zu. Anfangs liefert die Pumpe Wasser mit einer F&#xF6;rderrate ...

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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.

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.

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Finding Water Volume with Integration

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