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 Airflow

Airflow 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:

where dV/dt is the airflow rate in liters per second, and t is time in seconds.

Integration Using Substitution

To determine the volume V(t) of inhaled air at any time t, the airflow function is integrated:

Using the substitution u = 2πx/5, the integral simplifies to:

This integral evaluates to:

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.