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

Water flows into a large industrial storage tank, but the inflow rate is not constant. Initially, the pump delivers water at 5 m3/s.The water inflow is increasing at a constant rate of two cubic meters per second due to continuous system adjustments.This steady change makes the water inflow rate a linear function of time, where 2 is the slope showing the rate of increase, and 5 is the initial inflow rate.The goal is to find the total volume of water stored in the tank at any point in time. This requires indefinite integration, which finds the total volume of water by integrating the rate of water inflow over time.Instead of analyzing inflow at specific instants, integration provides a single function for the total volume of water with respect to time.Integrating the linear inflow rate introduces a quadratic term, which shows the accelerating growth of the water volume, while the linear term shows the constant growth from the initial inflow rate.Finally, the constant of integration is given by the volume function at t equals zero, which corresponds to the initial volume of water present in the tank.

indefinite integrals

The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:27.13%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19984/19984_Equation_1_editor_19984.svg" alt="Linear function, equation f(t)=2t+5, educational math concept, formula representation." data-altupdatedat="1775223722000" data-alt="19984_Equation_1.svg" /></figure>It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing the following integral:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:27.6%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19984/19984_Equation_2_editor_19984.svg" alt="Integration formula V(t)=&#8747;(2t+5)dt, calculus equation for educational reference." data-altupdatedat="1775223722000" data-alt="19984_Equation_2.svg" /></figure>When the power rule is applied to evaluate the integral, both quadratic and linear terms emerge, reflecting the changing rate of inflow:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:30.41%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19984/19984_Equation_3_editor_19984.svg" alt="Quadratic equation V(t)=t^2+5t+C, formula analysis, mathematical calculation." data-altupdatedat="1775223722000" data-alt="19984_Equation_3.svg" /></figure>The term t2 arises due to the increasing inflow rate, demonstrating that water accumulation follows a quadratic pattern over time. The term 5t represents the steady inflow component that exists from the beginning. The constant C accounts for the initial volume of water in the tank before the inflow process begins. Since integration determines the total volume by summing the instantaneous rate of change, this equation provides a comprehensive function that describes the total amount of water in the tank at any given moment.

Indefinite Integrals

The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the ...

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JoVE Core

Calculus

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Integrals

KEY TERMS AND CONCEPTS

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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