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

물이 대형 산업용 저장 탱크로 흐르지만, 유입 속도는 일정하지 않습니다. 초기에는 펌프가 5 m3/s의 속도로 물을 공급합니다.시스템 조정으로 인해 물 유입량은 초당 2입방미터의 일정한 속도로 증가하고 있습니다.이 꾸준한 변화로 인해 물 유입량은 시간에 대한 선형 함수가 되며, 2는 증가율을 나타내는 기울기, 5는 초기 유입량입니다.목표는 언제든 탱크에 저장된 총 물의 부피를 찾는 것입니다. 이는 시간에 따른 물 유입 속도를 적분하여 전체 물의 부피를 구하는 무한한 적분법을 필요로 합니다.특정 시점의 유입을 분석하는 대신, 적분은 시간에 따른 전체 물의 부피에 대한 단일 함수를 제공합니다.선형 유입 속도를 적분하면 수량의 증가를 나타내는 2차 항이 도입되고, 선형 항은 초기 유입량에서 일정한 증가를 나타냅니다.마지막으로, 적분 상수는 t 에서의 부피 함수가 0과 같아지며, 이는 탱크 내 초기 물의 부피에 해당합니다.

부정적분

&#xC800;&#xC7A5; &#xD0F1;&#xD06C;&#xB85C; &#xC720;&#xC785;&#xB418;&#xB294; &#xBB3C;&#xC758; &#xC720;&#xB7C9;&#xC740; &#xC77C;&#xC815;&#xD558;&#xC9C0; &#xC54A;&#xC73C;&#xBA70; &#xC2DC;&#xAC04;&#xC774; &#xC9C0;&#xB0A8;&#xC5D0; &#xB530;&#xB77C; &#xC99D;&#xAC00;&#xD569;&#xB2C8;&#xB2E4;. &#xCD08;&#xAE30;&#xC5D0;&#xB294; &#xD38C;&#xD504;&#xAC00; 5 L/min&#xC758; &#xC720;&#xB7C9;&#xC73C;&#xB85C; &#xBB3C;&#xC744; &#xACF5;&#xAE09;&#xD569;&#xB2C8;&#xB2E4;. &#xADF8;&#xB7EC;&#xB098; &#xC555;&#xB825;&#xC758; &#xC99D;&#xAC00;&#xB098; &#xC2DC;&#xC2A4;&#xD15C; &#xC870;&#xC815;&#xC73C;&#xB85C; &#xC778;&#xD574; &#xC720;&#xC785; &#xC720;&#xB7C9;&#xC740; &#xB9E4;&#xBD84; 2 L/min&#xC529; &#xC99D;&#xAC00;&#xD569;&#xB2C8;&#xB2E4;. &#xC774;&#xB7EC;&#xD55C; &#xC0C1;&#xD669;&#xC740; &#xB2E4;&#xC74C;&#xACFC; &#xAC19;&#xC740; &#xC120;&#xD615; &#xD568;&#xC218;&#xB85C; &#xC218;&#xD559;&#xC801;&#xC73C;&#xB85C; &#xB098;&#xD0C0;&#xB0BC; &#xC218; &#xC788;&#xC2B5;&#xB2C8;&#xB2E4;.\begin{equation}f(t) = 2t + 5\end{equation}&#xC2DC;&#xAC04;&#xC5D0; &#xB530;&#xB77C; &#xD0F1;&#xD06C;&#xC5D0; &#xCD94;&#xAC00;&#xB418;&#xB294; &#xCD1D; &#xBB3C;&#xC758; &#xCCB4;&#xC801;&#xC744; &#xAD6C;&#xD558;&#xAE30; &#xC704;&#xD574;&#xC11C;&#xB294; &#xC720;&#xC785; &#xC720;&#xB7C9; &#xD568;&#xC218;&#xB97C; &#xC801;&#xBD84;&#xD574;&#xC57C; &#xD569;&#xB2C8;&#xB2E4;. &#xCD1D; &#xBB3C;&#xC758; &#xCCB4;&#xC801; V(t)&#xB294; &#xB2E4;&#xC74C; &#xC801;&#xBD84;&#xC744; &#xC218;&#xD589;&#xD568;&#xC73C;&#xB85C;&#xC368; &#xC5BB;&#xC744; &#xC218; &#xC788;&#xC2B5;&#xB2C8;&#xB2E4;.\begin{equation}V(t) = \int (2t + 5) \, dt\end{equation}&#xC774; &#xC801;&#xBD84;&#xC744; &#xD3C9;&#xAC00;&#xD558;&#xAE30; &#xC704;&#xD574; &#xAC70;&#xB4ED;&#xC81C;&#xACF1; &#xBC95;&#xCE59;&#xC744; &#xC801;&#xC6A9;&#xD558;&#xBA74;, &#xC720;&#xC785; &#xC720;&#xB7C9;&#xC758; &#xBCC0;&#xD654; &#xD2B9;&#xC131;&#xC744; &#xBC18;&#xC601;&#xD558;&#xB294; &#xC774;&#xCC28;&#xD56D;&#xACFC; &#xC120;&#xD615;&#xD56D;&#xC774; &#xD568;&#xAED8; &#xB098;&#xD0C0;&#xB0A9;&#xB2C8;&#xB2E4;.\begin{equation}V(t) = t^2 + 5t + C\end{equation}t^2 &#xD56D;&#xC740; &#xC2DC;&#xAC04;&#xC774; &#xC9C0;&#xB0A8;&#xC5D0; &#xB530;&#xB77C; &#xC720;&#xC785; &#xC720;&#xB7C9;&#xC774; &#xC99D;&#xAC00;&#xD568;&#xC5D0; &#xB530;&#xB77C; &#xBC1C;&#xC0DD;&#xD558;&#xBA70;, &#xBB3C;&#xC758; &#xB204;&#xC801;&#xC774; &#xC2DC;&#xAC04;&#xC5D0; &#xB530;&#xB77C; &#xC774;&#xCC28;&#xD568;&#xC218;&#xC801;&#xC73C;&#xB85C; &#xC9C4;&#xD589;&#xB428;&#xC744; &#xB098;&#xD0C0;&#xB0C5;&#xB2C8;&#xB2E4;. 5t &#xD56D;&#xC740; &#xCC98;&#xC74C;&#xBD80;&#xD130; &#xC874;&#xC7AC;&#xD558;&#xB294; &#xC77C;&#xC815;&#xD55C; &#xC720;&#xC785; &#xC131;&#xBD84;&#xC744; &#xB098;&#xD0C0;&#xB0C5;&#xB2C8;&#xB2E4;. &#xC0C1;&#xC218; C&#xB294; &#xC720;&#xC785; &#xACFC;&#xC815;&#xC774; &#xC2DC;&#xC791;&#xB418;&#xAE30; &#xC804;&#xC5D0; &#xD0F1;&#xD06C;&#xC5D0; &#xC774;&#xBBF8; &#xC874;&#xC7AC;&#xD558;&#xB358; &#xBB3C;&#xC758; &#xCD08;&#xAE30; &#xCCB4;&#xC801;&#xC744; &#xC758;&#xBBF8;&#xD569;&#xB2C8;&#xB2E4;. &#xC801;&#xBD84;&#xC740; &#xC21C;&#xAC04;&#xC801;&#xC778; &#xBCC0;&#xD654;&#xC728;&#xC744; &#xB204;&#xC801;&#xD558;&#xC5EC; &#xCD1D;&#xB7C9;&#xC744; &#xAD6C;&#xD558;&#xB294; &#xACFC;&#xC815;&#xC774;&#xBBC0;&#xB85C;, &#xC774; &#xC2DD;&#xC740; &#xC784;&#xC758;&#xC758; &#xC2DC;&#xC810;&#xC5D0;&#xC11C; &#xD0F1;&#xD06C;&#xC5D0; &#xC800;&#xC7A5;&#xB41C; &#xCD1D; &#xBB3C;&#xC758; &#xCCB4;&#xC801;&#xC744; &#xC124;&#xBA85;&#xD558;&#xB294; &#xD3EC;&#xAD04;&#xC801;&#xC778; &#xD568;&#xC218;&#xB97C; &#xC81C;&#xC2DC;&#xD569;&#xB2C8;&#xB2E4;.

&#xC800;&#xC7A5; &#xD0F1;&#xD06C;&#xB85C; &#xC720;&#xC785;&#xB418;&#xB294; &#xBB3C;&#xC758; &#xC720;&#xB7C9;&#xC740; &#xC77C;&#xC815;&#xD558;&#xC9C0; ...

Education

교육

JoVE Core

Calculus

미적분학

Unit

단위

Integrals

적분

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Indefinite Integrals

무한적분

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

area problem

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Definite Integrals in Motion Analysis

Definite integrals can be used to find displacement from a velocity-time graph. A velocity function v(t) describes how motion changes over time. The definite integral of v(t) from a to b gives the net displacement between those times. It also represents the signed area under the velocity curve between the two points. Two useful properties make definite integrals easier to work with in motion problems. The additivity property says that if a continuous function is integrated over adjacent...

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The definite integral helps describe motion by showing how far an object travels over time. In this lesson, the key ideas are the Positivity Property and the Comparison Property of definite integrals. Both ideas connect integration to velocity and displacement in a way that is easy to interpret in real situations. The Positivity Property says that if an object’s velocity is nonnegative, then its total displacement over the time interval is also nonnegative. Nonnegative velocity means the...

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Accumulation and Rates in the Fundamental Theorem

The Fundamental Theorem of Calculus connects accumulation and rate of change. In this lesson, the idea is shown with a water tank that is filled by a pump. The pump rate changes over time, so the total water collected must be found by adding up the changing inflow. In water management, this kind of calculation matters for pressure control, operation timing, and system safety. The flow rate is given by a time-dependent function. To find how much water has entered the tank from the start to a...

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Definite Integrals with Antiderivatives

The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative. If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because...

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Integration in Breathing Cycle Modeling

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indefinite integrals

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...

net change theorem

Calculus for Reservoir Overflow Prediction

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Work Done on a Nonlinear Spring

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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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