1. Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 6 Accumulating Change: Limits of Sums and the Definite Integral

2. Copyright © by Houghton Mifflin Company, All rights reserved. Chapter 6 Key Concepts Results of ChangeResults of ChangeResults of ChangeResults of Change Approximating Area with RectanglesApproximating Area with RectanglesApproximating Area with RectanglesApproximating Area with Rectangles Limits of Sums of Rectangle AreasLimits of Sums of Rectangle AreasLimits of Sums of Rectangle AreasLimits of Sums of Rectangle Areas Introduction to Accumulation FunctionsIntroduction to Accumulation FunctionsIntroduction to Accumulation FunctionsIntroduction to Accumulation Functions Fundamental Theorem and AntiderivativesFundamental Theorem and AntiderivativesFundamental Theorem and AntiderivativesFundamental Theorem and Antiderivatives Definite IntegralsDefinite IntegralsDefinite IntegralsDefinite Integrals Improper IntegralsImproper IntegralsImproper IntegralsImproper Integrals

3. Copyright © by Houghton Mifflin Company, All rights reserved. Results of Change The accumulated change in a quantity is represented as the area or signed area of a region between the rate-of-change function for that quantity and the horizontal axis provided the function does not cross the horizontal axis.The accumulated change in a quantity is represented as the area or signed area of a region between the rate-of-change function for that quantity and the horizontal axis provided the function does not cross the horizontal axis.

4. Copyright © by Houghton Mifflin Company, All rights reserved. Results of Change: Example A water tank drains at a rate of r(t) = -2t gallons per minute t minutes after the water began draining. Determine the change in the volume of water in the tank during the first 4 minutes the tank was draining. 1/2 (4 minutes) (-8 gallons per minute) = -16 gallons The tank lost 16 gallons of water in the first 4 minutes.

5. Copyright © by Houghton Mifflin Company, All rights reserved. Results of Change: Exercise 6.1 #15 The rate of change of the population of North Dakota from 1970 through 1990 is modeled on the graph below. Find the area between the graph of p and the horizontal axis from 0 to 15 and interpret. (3870 people per year)(15 years) = 58,050 people From 1970 to 1985, the population of North Dakota increased by 58,050 people.

6. Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Area with Rectangles The sum of the areas of rectangles may be used to estimate the area beneath a curve.The sum of the areas of rectangles may be used to estimate the area beneath a curve.

7. Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Area: Example The absorption rate of a drug given in 20 mg doses may be modeled by r(x) = 1.708(0.845) x  g/mL/day when 0  x  20. Estimate the change in the drug concentration over the first twenty days.

8. Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Area: Example 2 4 6 8 10 12 14 16 18 20 Area of rectangle = height width Right endpoint of rectangle x Height of rectangle r(x) 1.220 0.871 0.622 0.444 0.317 0.226 0.162 0.115 0.082 0.059 2.439 1.742 1.244 0.888 0.634 0.453 0.323 0.231 0.165 0.118

9. Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Area: Example The sum of the areas of the rectangles is 8.235  g/mL. The drug concentration over the 20-day period increased by approximately 8.24  g/mL. From the graph, we can see that this is an underestimate.

10. Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Area: Exercise 6.1 #13 An cyclist is clocked by her coach at 10-minute intervals during an hour long ride. Estimate the distance traveled by using left and right rectangles. 22 18 20 23 15 17 12 0 10 20 30 40 50 60 Time (minutes) Speed (mph) left rectangle sum = (115 mph)(1/6 hour) = 19.17 miles

11. Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Area: Exercise 6.1 #13 An cyclist is clocked by her coach at 10-minute intervals during an hour long ride. Estimate the distance traveled by using left and right rectangles. 22 18 20 23 15 17 12 0 10 20 30 40 50 60 Time (minutes) Speed (mph) right rectangle sum = (105 mph)(1/6 hour) = 17.5 miles

12. Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Sums of Rectangle Areas Increasing the number of rectangles used to estimate the area between a curve and the horizontal axis increases the accuracy of the estimateIncreasing the number of rectangles used to estimate the area between a curve and the horizontal axis increases the accuracy of the estimate

13. Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Sums of Rectangle Areas Let f be a continuous or piecewise continuous non-negative function from a to b. The area of the region between the graph of f and the x-axis from a to b is given by the limit where x 1, x 2,..., x n are the midpoints of n subintervals of length  x = (b-a)/n between a and b. is called the definite integral of f from a to b.Let f be a continuous or piecewise continuous non-negative function from a to b. The area of the region between the graph of f and the x-axis from a to b is given by the limit where x 1, x 2,..., x n are the midpoints of n subintervals of length  x = (b-a)/n between a and b. is called the definite integral of f from a to b.

14. Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Sums: Example The flow rates of the Carson River can be modeled by f(x) = 18,225x 2 -135,334.3x + 2,881,542.9 ft 3 /hr where x is the number of hours after 11:45 a.m. Wednesday. Use the idea of limit of sums to estimate to the nearest ten thousand cubic feet the amount of water that flowed through the river from 11:45 a.m. on Wednesday to 7:45 a.m. on Thursday. We need to find the area between the curve and the x-axis from x = 0 to x = 20. We’ll use midpoint rectangles.

15. Copyright © by Houghton Mifflin Company, All rights reserved. 79,042,498 79,133,623 79,156,404 79,162,100 79,163,523 79,163,879 79,163,968 10 20 40 80 160 320 640 Number of Rectangles Approximate Area Limits of Sums: Example Approximately 79,160 thousand gallons of water flowed through the river in the 20-hour period. This estimate is accurate to the nearest 10,000 cubic feet.

16. Copyright © by Houghton Mifflin Company, All rights reserved. Limits of Sums: Exercise 6.2 #7 The rate of change of the weight of a mouse can be modeled by w(t) = 13.785t -1 grams per week where t is the age of the mouse in weeks and 1  x  15. Use the limit of sums to estimate 18.02742 17.94007 17.91799 17.91246 8 16 32 64 Number of Rectangles Approximate Area Between the third week and the eleventh week, the mouse gained about 17.9 grams. (Table values determined with technology.)

17. Copyright © by Houghton Mifflin Company, All rights reserved. Introduction to Accumulation Functions The accumulation function of a function f gives the accumulation of the area between the horizontal axis and the graph of f from a to x. The constant a is the starting value of the accumulation. The accumulation function is denoted byThe accumulation function of a function f gives the accumulation of the area between the horizontal axis and the graph of f from a to x. The constant a is the starting value of the accumulation. The accumulation function is denoted by

18. Copyright © by Houghton Mifflin Company, All rights reserved. Accumulation Functions: Example f(t) is the speed of a vehicle during a 10-second drive. The shaded area is the distance traveled over the first x seconds with x = 1, 2, 3, 4, and 5.

19. Copyright © by Houghton Mifflin Company, All rights reserved. Accumulation: Exercise 6.3 #19 Find a formula for the accumulation functions below and shade the area represented by it. A(x) = 6x A(x) = -2x

20. Copyright © by Houghton Mifflin Company, All rights reserved. Fundamental Theorem / Antiderivatives Let f be a function of x. A function F is called an antiderivative of f if F'(x) = f(x); that is, the derivative of F is f.Let f be a function of x. A function F is called an antiderivative of f if F'(x) = f(x); that is, the derivative of F is f. The Fundamental Theorem of CalculusThe Fundamental Theorem of Calculus –For any continuous function f with input t, the derivative of the accumulation function of f is the function f in terms of x. In symbols, we write

21. Copyright © by Houghton Mifflin Company, All rights reserved. Fundamental Theorem / Antiderivatives Antiderivative FormulasAntiderivative Formulas –Simple Power Rule for Antiderivatives –Constant Multiplier Rule for Antiderivatives –Sum Rule

22. Copyright © by Houghton Mifflin Company, All rights reserved. Fundamental Theorem / Antiderivatives Antiderivative FormulasAntiderivative Formulas –Natural Log Rule –e ax Rule –Exponential Rule

23. Copyright © by Houghton Mifflin Company, All rights reserved. Fundamental Thm. / Antiderivatives: Example An African country has an increasing population but a declining birth rate. The rate of change in the number of babies born each year is given by b(t) = 87,000 - 1600t births per year t years from the end of this year. The number of babies born this year is 1,185,800. Find a function describing the number of births each year t years from now.

24. Copyright © by Houghton Mifflin Company, All rights reserved. Antiderivatives : Exercise 6.4 #39 The rate of change of the growth rate of corn during the first two months after it is planted may be modeled by g'(d) = -0.00036d - 0.018 inches per day squared where d is the number of days after sprouting. Find the function for sweet corn growth rate given it was growing 1.95 inches per day on the day it sprouted.

25. Copyright © by Houghton Mifflin Company, All rights reserved. Definite Integrals If f is a continuous function from a to b and F is any antiderivative of f, thenIf f is a continuous function from a to b and F is any antiderivative of f, then The definite integral from a to b represents the total change in F over the interval a to bThe definite integral from a to b represents the total change in F over the interval a to b

26. Copyright © by Houghton Mifflin Company, All rights reserved. Definite Integrals: Example The marginal cost for toaster ovens may be modeled by C'(x) = 47.638(0.0072) x dollars per day. If the manufacturer wishes to increase production from 300 to 500 ovens per day, how much will the total cost change?

27. Copyright © by Houghton Mifflin Company, All rights reserved. Definite Integrals: Exercise 6.5 #22 The air speed of a small airplane during the first 25 seconds of takeoff and flight can be modeled by t minutes after takeoff. Calculate and interpret the following definite integral. The plane traveled 0.200175 miles in the first 18 seconds (0.005 hours) of takeoff and flight.

28. Copyright © by Houghton Mifflin Company, All rights reserved. Improper Integrals Improper integrals are integrals of one of the following three forms:Improper integrals are integrals of one of the following three forms:

29. Copyright © by Houghton Mifflin Company, All rights reserved. Improper Integrals: Example

30. Copyright © by Houghton Mifflin Company, All rights reserved. Improper Integrals: Exercise 6.6 #23