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

Breathing is a cyclic process, with each full respiratory cycle lasting about 5 seconds.The goal is to find the volume of inhaled air in the lungs at any time t during this cycle.The rate of airflow changes over time and can be modeled using a sinusoidal function. It shows that airflow peaks during mid-inhalation and reverses during exhalation, with a maximum rate of about 0.5 L/s.The infinitesimal volume dV over a small time dt accumulates to the total inhaled volume by integrating the airflow rate from 0 to t.The substitution method simplifies the integral by setting the variable u as 2&#960;x over 5. The differential dx is expressed in terms of du to match the new variable. The limits of integration also change to match the new variable, ranging from 0 to 2&#960;t over 5.The integral becomes a standard sine form, whose integral is negative cosine, evaluated using the limits of integration.This gives the volume of inhaled air in liters as a function of time during the breathing cycle, reaching its maximum during inhalation.This model includes both direction and rate of airflow, giving a clear view of respiratory volume over time.

application of integration: problem solving

The process of breathing involves the periodic intake and expulsion of air, known as the respiratory cycle, which typically lasts about five seconds. Modeling the volume of air inhaled into the lungs as a function of time provides insight into both the dynamics and efficiency of pulmonary ventilation. This volume is determined by integrating the airflow rate over time, which captures the cumulative effect of air entering the lungs.Sinusoidal Model of AirflowAirflow during respiration is not constant; it varies over time and is most accurately represented by a sinusoidal function. In a normal breathing cycle, the airflow rate reaches a maximum of approximately 0.5 liters per second at the midpoint of inhalation and decreases symmetrically toward zero at the start and end of each phase. This periodic behavior can be modeled by the function:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:30.88%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19983/19983_Resized_PT_Equation_1_editor_19983.svg" alt="Differential equation diagram: dV/dt=0.5sin(2&#960;t/5), dynamic system analysis." data-altupdatedat="1775223722000" data-alt="19983_Resized_PT_Equation_1.svg" /></figure>where dV/dt is the airflow rate in liters per second, and t is time in seconds.Integration Using SubstitutionTo determine the volume V(t) of inhaled air at any time t, the airflow function is integrated:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:28.04%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19983/19983_Resized_PT_Equation_2_editor_19983.svg" alt="Integrating sinusoidal function equation; mathematical analysis; calculus concept." data-altupdatedat="1775223722000" data-alt="19983_Resized_PT_Equation_2.svg" /></figure>Using the substitution u = 2&#960;x/5, the integral simplifies to:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19983/19983_Resized_equation_3_editor_19983.svg" alt="Voltage function equation, integral calculus formula; mathematical expression, educational diagram." data-altupdatedat="1775223722000" data-alt="19983_Resized_equation_3.svg" /></figure>This integral evaluates to:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:29.01%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19983/19983_Resized_PT_Equation_4_editor_19983.svg" alt="Static equilibrium; V(t)=-5/4&#960;[cos(2&#960;t/5)-1]; equation; signal analysis; harmonic oscillation." data-altupdatedat="1775223722000" data-alt="19983_Resized_PT_Equation_4.svg" /></figure>This function describes the volume of inhaled air over time, capturing both the magnitude and direction of airflow. The model peaks at mid-inhalation, reflecting the physiological behavior of the lungs during a typical breath cycle.

Application of Integration: Problem Solving

The process of breathing involves the periodic intake and expulsion of air, known as the respiratory cycle, which typically lasts about five seconds. ...

application-of-integration-problem-solving

Education

JoVE Core

Calculus

Unit

Integrals

KEY TERMS AND CONCEPTS

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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