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

La respiration est un processus cyclique, chaque cycle respiratoire complet durant environ 5 secondes.L’objectif est de déterminer le volume d’air inhalé dans les poumons à tout moment durant ce cycle.Le taux de débit d’air change au fil du temps et peut être modélisé à l’aide d’une fonction sinusoïdale. Il montre que le débit d’air atteint son pic pendant l’inspiration et s’inverse à l’expiration, avec un débit maximal d’environ 0,5 L/s.Le volume infinitésimal dV , sur un court temps , s’accumule jusqu’au volume total inhalé en intégrant le débit d’air de 0 à t.La méthode de substitution simplifie l’intégrale en fixant la variable u à 2πx sur 5. Le différentiel dx est exprimé en termes de du pour correspondre à la nouvelle variable. Les limites d’intégration changent également pour correspondre à la nouvelle variable, allant de 0 à 2π sur 5.L’intégrale devient une forme sinusoïdal standard, dont l’intégrale est un cosinus négatif, évaluée en utilisant les limites d’intégration.Cela donne le volume d’air inhalé en litres en fonction du temps pendant le cycle respiratoire, atteignant son maximum lors de l’inspiration.Ce modèle inclut à la fois la direction et la vitesse de l’écoulement d’air, offrant une vue claire du volume respiratoire au fil du temps.

application de l’intégration : résolution de problèmes

La respiration est un processus p&#xE9;riodique d&#x2019;entr&#xE9;e et de sortie de l&#x2019;air, appel&#xE9; cycle respiratoire, qui dure g&#xE9;n&#xE9;ralement environ cinq s. La mod&#xE9;lisation du volume d&#x2019;air inhal&#xE9; dans les poumons en fonction du temps &#xE9;claire &#xE0; la fois la dynamique et l&#x2019;efficacit&#xE9; de la ventilation pulmonaire. Ce volume est obtenu en int&#xE9;grant le d&#xE9;bit d&#x2019;air au cours du temps, ce qui rend compte de l&#x2019;effet cumulatif de l&#x2019;air entrant dans les poumons.Mod&#xE8;le sinuso&#xEF;dal du d&#xE9;bit d&#x2019;airLe d&#xE9;bit d&#x2019;air pendant la respiration n&#x2019;est pas constant ; il varie dans le temps et se mod&#xE9;lise le plus fid&#xE8;lement par une fonction sinuso&#xEF;dale. Lors d&#x2019;un cycle respiratoire normal, le d&#xE9;bit d&#x2019;air atteint un maximum d&#x2019;environ 0,5 L/s au milieu de l&#x2019;inspiration et diminue sym&#xE9;triquement vers z&#xE9;ro au d&#xE9;but et &#xE0; la fin de chaque phase. Ce comportement p&#xE9;riodique peut &#xEA;tre mod&#xE9;lis&#xE9; par la fonction :\begin{equation*}\jfrac{dV}{dt} = 0.5 \sin\liparens{\jfrac{2\pi t}{5}}\end{equation*}o&#xF9; dV/dt est le d&#xE9;bit d&#x2019;air en litres par seconde, et t le temps en secondes.Int&#xE9;gration par substitutionPour calculer le volume V(t) d&#x2019;air inhal&#xE9; &#xE0; l&#x2019;instant t, on int&#xE8;gre la fonction donnant le d&#xE9;bit d&#x2019;air :\begin{equation*}V(t) = \int_0^t 0.5 \sin\liparens{\jfrac{2\pi x}{5}}dx\end{equation*}En utilisant la substitution u = 2&#x3C0;x/5, l&#x2019;int&#xE9;grale se simplifie en :\begin{equation*}V(t) = \jfrac{5}{2\pi} \int_0^{\jfrac{2\pi t}{5}} 0.5 \sin(u) du\end{equation*}Cette int&#xE9;grale vaut :\begin{equation*}V(t) = -\jfrac{5}{4\pi} \left[\cos\left(\jfrac{2\pi t}{5}\right) - 1\right]\end{equation*}Cette fonction d&#xE9;crit le volume d&#x2019;air inhal&#xE9; au cours du temps, en tenant compte &#xE0; la fois de l&#x2019;amplitude et du sens du d&#xE9;bit d&#x2019;air. Le mod&#xE8;le atteint un maximum en milieu d&#x2019;inspiration, ce qui refl&#xE8;te le comportement physiologique des poumons lors d&#x2019;un cycle respiratoire typique.

Application de l’intégration : Résolution de problèmes

La respiration est un processus p&#xE9;riodique d&#x2019;entr&#xE9;e et de sortie de l&#x2019;air, appel&#xE9; cycle respiratoire, qui dure ...

application-of-integration-problem-solving

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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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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...

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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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