Use the result of Exercise 4.5 .57 to compute the average volume of inhaled air in the lungs in

Use the result of Exercise 4.5 .57 to compute the average...

Key Concepts

Average Value of a Function

This concept involves finding the mean value of a continuous quantity by integrating a function over a certain interval and then dividing by the length of that interval. It is crucial when you need to determine a representative value from a continuously varying set of data, such as the volume of air during a respiratory cycle.

Definite Integration

Definite integration is the process of calculating the total accumulation of a function's values over a specific interval. In contexts like calculating the total or average volume, integration combines infinitesimal contributions to yield the overall physical quantity, which is a foundational tool in calculus.

Application of Calculus to Physiological Phenomena

Applying calculus to physiology allows for modeling and analysis of complex biological processes quantitatively. For instance, by representing the volume of air inhaled as a function of time during a respiratory cycle, calculus techniques can be used to compute average values that are critical for understanding and predicting respiratory performance.

Transcript

00:03 Okay, so we are doing this problem that asked to find the average value of inhaled air in the lungs in one respiratory cycle using a problem from exercise 4 .5, number 57.

00:23 So in that problem from section 4 .5, the answer for the...

00:36 Volume of air function is 5 over 4 pi times 1 minus cosine of 2 pi 2 pi t over 5.

01:00 Okay, so that was the function for the volume of air in terms of t, which is the number of seconds.

01:13 So in this problem, it asks for the average volume of inhaled air in one respiratory cycle.

01:20 So from number 47, one respiratory cycle is 5 seconds.

01:26 So one respiratory cycles equals 5 seconds.

01:32 Okay, so to find the average volume of air in one cycle, which is 5 second, we will use the average value of function formula for any function f.

01:45 It's 1 over b minus a times the integral of a to b of f of x, dx.

01:51 So we know that it's 5 seconds, so a to b is 0 to 5.

01:58 So our integral is going to be 1 over 5 minus 0 of the volume formula.

02:14 Since the problem asked for the average volume.

02:20 So times one times the volume formula, so v of t d t.

02:31 Okay, so that becomes 1 over 5, 0 to 5 times 5 over 4 pi times 1 minus cosine 2 pi t over 5 d t.

02:51 Okay, so simplifying this integral before we integrate, we have 1 over 4 pi from 0 to 5 of 1 minus cosine 2 pi 2 pi t over 5 d t.

03:34 Okay, so now we can integrate.

03:38 And you can integrate by u substitution...