#### To determine

**To compute:**

The average volume of the inhaled air in the lungs in one respiratory cycle.

#### Answer

Vtave=54πL

#### Explanation

**1) Concept:**

Use the mean value theorem for integrals and find theaverage volume of the inhaled air in the lungs in one respiratory cycle

**2) Mean Value Theorem for Integrals**

If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-a∫abfx dx

that is,

∫abfx dx=fcb-a

**3) Given:**

The result of the exercise 4.5.57, the volume of the inhaled air in the lungs at the time t is

Vt=54π1-cos2πt5 liters for 0≤t≤5

**4) Calculation:**

By using the mean value theorem for integrals:

The average volume of the inhaled air in the lungs in one respiratory cycle is

Vtave=15-0∫0554π1-cos2πt5 dt

=15∫0554π1-cos2πt5 dt

=14π∫051-cos2πt5 dt

=14π∫05 dt-∫05cos2πt5 dt

Integrating,

=14πt05-2π5sin2πt505

=14π5-2π5sin2π

Vtave=54π

**Conclusion:**

The average volume of the inhaled air in the lungs in one respiratory cycle is 54πL.