Section 5.4 Interpretations of The Definite Integral But first two exercises from section 5.1, a review exercise, and questions on the 5.2-3 homework. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Exercise 5.1 #14 The figure shows the rate of change of a fish population. Estimate the total change in the population. Lower sum = 10 + 21 + 10 × 4 = 164 Upper sum = 21 + 22 + 21 × 4 = 256 Left sum = 10 + 21 + 21 × 4 = 208 Right sum = 21 + 21 + 10 × 4 = 208 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. Speed (mph) Fuel efficiency (mpg) miles/hr × miles/gal 10 15 20 18 30 21 40 23 50 24 60 25 hr = gal Extremely Low = 10 1 26 2 ≈ 0.19 gal Extremely High = 10 20 30 40 50 60 1 + + + + + 18 21 23 24 25 26 12 ≈ 0.73 gal Very Low = Low = 15 25 35 45 55 65 1 + + + + + 18 21 23 24 25 26 12 ≈ 0.84 gal High = 15 25 35 45 55 65 1 + + + + + 15 18 21 23 24 25 12 ≈ 0.91 gal Very High = 20 30 40 50 60 70 1 + + + + + 15 18 21 23 24 25 12 ≈ 1.03 gal 70 1 15 2 ≈ 2.33 gal 70 26 Temperature Example Suppose π(π‘) is the outdoor temperature in °C at π‘ hours after midnight. The table provides known values of π. 0 3 8 5 10 7 12 10 18 2 24 -6 1. Estimate π′(10) and provide an interpretation. 10−5 12−8 = 1.25 °C/hr is the rate at which the temperature is changing at 10:00am. 2. Estimate π′′(10) and provide an interpretation. 1.5−1 11−9 = 0.25 °C/hr2 is the acceleration of the temperature at 10:00am. 3. Estimate the average temperature during the 24-hour day. (4 × 8 + 2 × 6 + 8.5 × 2 + 6 × 6 − 4 × 6)/24 ≈ 3.0 °C 4. Estimate the area between the graph of π¦ = π(π‘) and the π‘-axis. 4 × 8 + 2 × 6 + 8.5 × 2 + 6 × 6 + 1 × 2 + 3 × 4 ≈ 111 °C-hr 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? 10 ο² r ο¨t ο©dt 0 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 If the units for π(π₯) are feet per minute and the units of π₯ are feet, then what are the units of the following. 5 ο² f ο¨x ο©dx 0 Feet squared per minute. What might π(π₯) and the integral represent? π(π₯) could represent the rate at which the sides of a square are changing when the side has length π₯. 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 Suppose π(π‘) is the rate of change of the population of a city, in people per year, at time π‘ years since the start of 1990. If the population of the city is 5000 people at the start of 1990, give an expression for the population today. 13.88 5000 + π π‘ ππ‘ 0 If π π‘ = 60 π‘, then what is the population today? The Mathematica command 5000 + Integrate[60 Sqrt[π‘ , π‘, 0,13.88 yields 7068. 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 < 12, we have f (t) > g(t), so the number of sick people is increasing. For t > 12, we have f (t) < g(t), so the number of sick people is decreasing. Since the area between the curves is greater for t > 12 than the area between the curves for t < 12, the decrease is larger than the increase. The answer is (b). Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved.