Name: integration in breathing cycle modeling
Uploaded: 2026-01-12T12:24:28.000Z
Duration: 1 min 30 s
Channel: JoVE

호흡은 주기적인 과정으로, 각 완전한 호흡 주기는 약 5초간 지속됩니다.목표는 이 주기 중 언제든지 t에서 폐에 흡입된 공기의 부피를 찾는 것입니다.공기 흐름 속도는 시간에 따라 변하며, 사인파 함수를 사용해 모델링할 수 있습니다. 이 분석은 공기 흐름이 흡입 중 최고조에 달하고 내쉴 때 역전하며, 최대 속도는 약 0.5 L/s임을 보여줍니다.짧은 시간 dt 동안 무한히 작은 부피 dV는 0에서 t까지 공기흐름 속도를 적분하여 총 흡입 부피로 합적됩니다.치환법은 변수 u 를 2πx 5로 설정하여 적분을 단순화합니다. 미분 dx 는 새로운 변수와 일치하기 위해 du 단위로 표현됩니다. 적분의 한계도 새로운 변수에 맞게 변화하며, 0에서 5중 2 πt 까지 범위가 있습니다.적분은 표준 사인 형식이 되며, 적분은 음의 코사인이며, 적분의 극한을 사용하여 평가한다.이는 호흡 주기 동안 시간에 따른 흡입 공기의 부피(리터 단위)를 제공하며, 흡입 시 최대치에 도달합니다.이 모델은 공기 흐름의 방향과 속도를 모두 포함하여 시간에 따른 호흡량을 명확히 파악할 수 있습니다.

적분의 응용: 문제 풀이

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&#xD638;&#xD761; &#xACFC;&#xC815;&#xC740; &#xACF5;&#xAE30;&#xB97C; &#xC8FC;&#xAE30;&#xC801;&#xC73C;&#xB85C; &#xD761;&#xC785;&#xD558;&#xACE0; ...

application-of-integration-problem-solving

Education

교육

JoVE Core

Calculus

미적분학

Unit

단위

Integrals

적분

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Application of Integration: Problem Solving

통합의 적용: 문제 해결

integration in breathing cycle modeling

Integration can be used to model the volume of air inhaled during breathing. The respiratory cycle is the repeating intake and expulsion of air, and it usually lasts about five seconds. By treating airflow as a function of time, students can see how the amount of air in the lungs changes during a breath.

Airflow during respiration is not constant. It changes in a smooth, wave-like pattern, so a sinusoidal function is a good model. In a normal breathing cycle, the airflow rate reaches a maximum of about 0.5 liters per second at the midpoint of inhalation. It then falls symmetrically to zero at the start and end of each phase.

To find the inhaled volume V(t), the airflow rate is integrated over time. The model uses dV/dt in liters per second and time t in seconds. With the substitution u = 2πx/5, the integral becomes easier to solve. The result gives a function for the volume of inhaled air over time.

This integral describes how much air enters the lungs as breathing continues. It also captures both the size and direction of airflow. The volume curve peaks at mid-inhalation, which matches the normal pattern of lung movement during a typical breath cycle.

Integration in Breathing Cycle Modeling

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key terms and concepts

See how integration models inhaled air in the breathing cycle using a sinusoidal airflow function, substitution, and a volume curve for lung ventilation.

Integration in Breathing Cycle Modeling | 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

Integration in Breathing Cycle Modeling

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