Application of Integration: Problem Solving

Integration accumulates the infinitesimal volume changes over time to find total inhaled air. By integrating the airflow rate function from time 0 to t, you capture the cumulative effect of air entering the lungs throughout the respiratory cycle. This approach reveals how volume changes dynamically during inhalation and exhalation.

Airflow during breathing is not constant; it varies periodically over time. A sinusoidal function accurately represents this periodic behavior, peaking at approximately 0.5 liters per second during mid-inhalation and decreasing symmetrically toward zero at the start and end of each phase. This captures the natural oscillating pattern of the respiratory cycle.

The substitution rule applied to definite integrals simplifies complex expressions by introducing a new variable. Setting u = 2πx/5 transforms the integral into a standard sine form, making it easier to evaluate. The differential dx is rewritten in terms of du, and integration limits are adjusted to match the new variable.

When substituting u = 2πx/5, the original limits from 0 to t must be converted to the new variable. The lower limit 0 remains 0, while the upper limit t becomes 2πt/5. These adjusted limits ensure the integral evaluates over the same time interval in the new variable space.

The antiderivative of the sine function is negative cosine. After substitution, the integral of sin(u) evaluates to -cos(u), which is then evaluated using the adjusted limits of integration. This produces the volume function V(t) that describes inhaled air over time during the respiratory cycle.

The volume function V(t) captures both the magnitude and direction of airflow, showing how much air accumulates in the lungs at any time t. It peaks during mid-inhalation, reflecting physiological breathing behavior, and demonstrates how the cumulative effect of varying airflow rates produces the total inhaled volume throughout the five-second respiratory cycle.

Continuous modeling with integration captures the dynamic nature of respiration, where airflow varies throughout the cycle. Using constant values would ignore the physiological reality that airflow peaks mid-inhalation and decreases toward zero. Integration of the sinusoidal function provides an accurate, detailed view of how respiratory volume changes over time.