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

呼吸は周期的なプロセスで、各完全な呼吸サイクルは約5秒間続きます。目的は、この サイクル中の任意の 時刻tにおける肺に吸入された空気の量を求めることです。気流の速度は時間とともに変化し、正弦関数を用いてモデル化できます。この研究では、吸気中に空気のピークが現れ、呼気時に逆転し、最大流量は約0.5 L/sであることが示されています。微小体積 dV は、 短時間dt で空気流量を0から tに積分することで総吸気体積に蓄積されます。置換法は変数 u を2πx ×5として設定することで積分を簡素化します。微分 dx は新しい変数に合わせるために du で表されます。積分の極限も新しい変数に合わせて変化し、5に対して0から2πt の範囲で変化します。積分は標準的な正弦形式となり、積分は負の余弦となり、積分の極限を用いて評価されます。これにより、呼吸サイクル中の時間の関数として吸気された空気の体積(リットル単位)が示され、吸気時に最大になります。このモデルは気流の方向と速度の両方を含み、時間経過による呼吸量を明確に示します。

積分の応用：問題解法

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application-of-integration-problem-solving

Education

JoVE Core

Calculus

微積分学

Unit

単位

Integrals

積分

KEY TERMS AND CONCEPTS

主要用語と概念

Application of Integration: Problem Solving

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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