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

呼吸是一个周期性过程，每个完整呼吸周期大约持续5秒。目标是在这个周期中 任何时间t 内，求出肺部吸入的空气体积。气流速率随时间变化，可以用正弦函数进行建模。研究显示，气流在吸气中达到峰值，呼气时反转，最大流量约为0.5升/秒。在短时间dt内，无穷小体积dV通过从0积分到t的气流速率累积到总吸入体积。代换方法通过将变量 u 设为2πx 除以5来简化积分。微分 dx 用 du 表示以匹配新变量。积分的极限也会根据新变量的变化变化，范围范围为0到2πt ，除以5。积分成为标准正弦型，其积分为负余弦，利用积分极限进行计算。这给出了吸入空气体积（单位为升）随呼吸周期中时间变化的量，达到吸气时的最大值。该模型包含气流方向和速率，清晰地反映了随时间变化的呼吸体积。

积分的应用：问题解决

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application-of-integration-problem-solving

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

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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