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

水流入一个大型工业储水罐，但流入速率并不恒定。最初，泵以 5 m3/s 的速率供水。由于持续的系统调整，水流入量正以每秒两立方米的恒定速率增加。这种稳定变化使得水流流入速率成为时间的线性函数，其中 2 是表示增加速率的斜率，5 是初始流入速率。目标是求出在任意时刻水箱中储存的水的总体积。这需要通过不定积分来实现，即通过对水的流入速率随时间进行积分，从而求得水的总体积。与在特定时刻分析流入量不同，积分提供了一个关于时间的单一函数，用于表示水的总体积。对线性流入速率进行积分会引入一个二次项，该二次项表明水体积呈加速增长，而线性项则反映由初始流入速率带来的恒定增长。最后，积分常数由 t 等于零时的体积函数给出，这对应于水箱中水的初始体积。

不定积分

Indefinite Integrals

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Integrals

积分

KEY TERMS AND CONCEPTS

关键术语与概念

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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