Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Section 5.4 Interpretations of The Definite Integral But first two exercises from section 5.1, a review exercise, and questions on the homework.

Exercise 5.1 #14 The figure shows the rate of change of a fish population. Estimate the total change in the population. Either the average of the lower and upper sums, 210 fish, or the average of the left and right sums, 208 fish, provides a good estimate for the total change in the fish population.

Exercise 5.1 #20 A car speeds up at a constant rate from 10 to 70 mph over a period of half an hour. Its fuel efficiency (in miles per gallon) increases with speed; values are in the table. Make lower and upper estimates of the quantity of fuel used during the half hour.

Temperature Example

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Meaning of the Integral

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Heart Example Suppose r(t) is the rate at which the heart is pumping blood in liters per second and t is the time in seconds. What does the following mean? The amount of blood in liters pumped by the heart over the first ten seconds.

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Feet Example The integral could represent the rate at which some area has changed as some length has gone from 0 to 5 feet.

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Population Example

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Flu Example In Figure 5.11, the function f (t) gives the rate at which healthy people become sick with the flu, and g(t) is the rate at which they recover. Which of the graphs (a) – (d) could represent the number of people sick with the flu during a 30-day period? For t g(t), so the number of sick people is increasing. For t > 12, we have f (t) 12 than the area between the curves for t < 12, the decrease is larger than the increase. The answer is (b).