MA 126, Day 14 - Introduction to Integration

Transcript

1. How do we measure distance traveled? Summation Notation The Definite Integral Introduction to Integration Benjamin Katz-Moses November 15, 2012 Colorado Colorado 1 / 14 Introduction to Integration
2. How do we measure distance traveled? Summation Notation The Definite Integral Outline 1 How do we measure distance traveled? 2 / 14 Introduction to Integration
3. How do we measure distance traveled? Summation Notation The Definite Integral Outline 1 How do we measure distance traveled? 2 Summation Notation 2 / 14 Introduction to Integration
4. How do we measure distance traveled? Summation Notation The Definite Integral Outline 1 How do we measure distance traveled? 2 Summation Notation 3 The Definite Integral 2 / 14 Introduction to Integration
5. How do we measure distance traveled? Summation Notation The Definite Integral A Simple Example Example If you drive 60mph for two hours, how far did you travel?
6. How do we measure distance traveled? Summation Notation The Definite Integral A Simple Example Example If you drive 60mph for two hours, how far did you travel? Distance = Rate × Time
7. How do we measure distance traveled? Summation Notation The Definite Integral A Simple Example Example If you drive 60mph for two hours, how far did you travel? Distance = Rate × Time = (60 miles/hr) · (2 hr) = 60 miles 3 / 14 Introduction to Integration
8. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 The following data is gathered as a small plane travels down the runway toward takeoff. t (sec) 0 2 4 6 8 10 v(t) (ft/sec) 0 99 140 171.5 198 221.4 (a) How far did the plane travel in the 10 second period? 4 / 14 Introduction to Integration
9. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 The following data is gathered as a small plane travels down the runway toward takeoff. t (sec) 0 2 4 6 8 10 v(t) (ft/sec) 0 99 140 171.5 198 221.4 (a) How far did the plane travel in the 10 second period? Underestimate (0 + 99 + 140 + 171.5 + 198)(2) = 1217 ft 4 / 14 Introduction to Integration
10. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 The following data is gathered as a small plane travels down the runway toward takeoff. t (sec) 0 2 4 6 8 10 v(t) (ft/sec) 0 99 140 171.5 198 221.4 (a) How far did the plane travel in the 10 second period? Overestimate (99 + 140 + 171.5 + 198 + 221.4)(2) = 1659.8 ft 4 / 14 Introduction to Integration
11. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 (b) Illustrate your estimates in (a) with rectangles on the graph below. What does the area of each rectangle represent? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • 4 / 14 Introduction to Integration
12. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 (b) Illustrate your estimates in (a) with rectangles on the graph below. What does the area of each rectangle represent? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • 4 / 14 Introduction to Integration
13. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 (b) Illustrate your estimates in (a) with rectangles on the graph below. What does the area of each rectangle represent? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • 4 / 14 Introduction to Integration
14. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 (c) What is the maximum possible error in either of the estimates you found in part (a)? 5 / 14 Introduction to Integration
15. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 (c) What is the maximum possible error in either of the estimates you found in part (a)? The maximum possible error in is given by the difference between the overestimate and the underestimate. 1659.8 − 1217 = 442.8ft 5 / 14 Introduction to Integration
16. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #1 (c) What is the maximum possible error in either of the estimates you found in part (a)? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • 5 / 14 Introduction to Integration
17. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 Suppose that the same small plane as in Example 1 is traveling toward takeoff, but that now, we are given the velocity of the plane every second (as shown in the table below). t (sec) 0 1 2 3 4 5 6 7 8 9 10 v(t) (ft/sec) 0 75 99 125 140 162 171.5 182 198 215 221 (a) How far did the plane travel in the 10 second period? 6 / 14 Introduction to Integration
18. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 Suppose that the same small plane as in Example 1 is traveling toward takeoff, but that now, we are given the velocity of the plane every second (as shown in the table below). t (sec) 0 1 2 3 4 5 6 7 8 9 10 v(t) (ft/sec) 0 75 99 125 140 162 171.5 182 198 215 221 (a) How far did the plane travel in the 10 second period? Underestimate (0+75+99+125+140+162+171.5+182+198+215)(1) = 1367.5 ft 6 / 14 Introduction to Integration
19. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 Suppose that the same small plane as in Example 1 is traveling toward takeoff, but that now, we are given the velocity of the plane every second (as shown in the table below). t (sec) 0 1 2 3 4 5 6 7 8 9 10 v(t) (ft/sec) 0 75 99 125 140 162 171.5 182 198 215 221 (a) How far did the plane travel in the 10 second period? Overestimate (75+99+125+140+162+171.5+182+198+215+221.4)(1) = 1588.9 ft 6 / 14 Introduction to Integration
20. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 (b) Illustrate your estimates in (a) with rectangles on the graph below. What does the area of each rectangle represent? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • • • • • • 6 / 14 Introduction to Integration
21. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 (b) Illustrate your estimates in (a) with rectangles on the graph below. What does the area of each rectangle represent? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • • • • • • 6 / 14 Introduction to Integration
22. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 (b) Illustrate your estimates in (a) with rectangles on the graph below. What does the area of each rectangle represent? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • • • • • • 6 / 14 Introduction to Integration
23. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 (c) What is the maximum possible error in either of the estimates you found in part (a)? 7 / 14 Introduction to Integration
24. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 (c) What is the maximum possible error in either of the estimates you found in part (a)? The maximum possible error in is given by the difference between the overestimate and the underestimate. 1588.9 − 1367.5 = 221.4ft 7 / 14 Introduction to Integration
25. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #2 (c) What is the maximum possible error in either of the estimates you found in part (a)? v(t) t 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 225 • • • • • • • • • • • 7 / 14 Introduction to Integration
26. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check A sprinter practices by running various distances back and forth in a straight line in a gym. Her velocity at t seconds is given by the function v(t). What does 60 0 |v(t)|dt represent? 1 The total distance the sprinter ran in one minute 2 The sprinter’s average velocity in one minute 3 The sprinter’s distance from the starting point after one minute 4 None of the above 8 / 14 Introduction to Integration
27. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check A sprinter practices by running various distances back and forth in a straight line in a gym. Her velocity at t seconds is given by the function v(t). What does 60 0 |v(t)|dt represent? 1 The total distance the sprinter ran in one minute 2 The sprinter’s average velocity in one minute 3 The sprinter’s distance from the starting point after one minute 4 None of the above 8 / 14 Introduction to Integration
28. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check A sprinter practices by running various distances back and forth in a straight line in a gym. Her velocity at t seconds is given by the function v(t). What does 60 0 |v(t)|dt represent? 1 The total distance the sprinter ran in one minute 2 The sprinter’s average velocity in one minute 3 The sprinter’s distance from the starting point after one minute 4 None of the above The integral 60 0 v(t) dt represents the total displacement of the runner relative to her starting position. 8 / 14 Introduction to Integration
29. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #4 Calculate the sums 1 4 k=2 k3 = 2 5 i=3 1 i − 1 = 3 5 j=1 9 = 9 / 14 Introduction to Integration
30. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #4 Calculate the sums 1 4 k=2 k3 = 23 + 33 + 43 = 99 2 5 i=3 1 i − 1 = 3 5 j=1 9 = 9 / 14 Introduction to Integration
31. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #4 Calculate the sums 1 4 k=2 k3 = 23 + 33 + 43 = 99 2 5 i=3 1 i − 1 = 1 3 − 1 + 1 4 − 1 + 1 5 − 1 = 13 12 3 5 j=1 9 = 9 / 14 Introduction to Integration
32. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #4 Calculate the sums 1 4 k=2 k3 = 23 + 33 + 43 = 99 2 5 i=3 1 i − 1 = 1 3 − 1 + 1 4 − 1 + 1 5 − 1 = 13 12 3 5 j=1 9 = 9 + 9 + 9 + 9 + 9 = 45 9 / 14 Introduction to Integration
33. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check True or False. If a piece of string has been chopped into n small pieces and the ith piece is ∆xi inches long, then the total length of the string is exactly n i=1 ∆xi . 10 / 14 Introduction to Integration
34. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check True or False. If a piece of string has been chopped into n small pieces and the ith piece is ∆xi inches long, then the total length of the string is exactly n i=1 ∆xi . 10 / 14 Introduction to Integration
35. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check True or False. If a piece of string has been chopped into n small pieces and the ith piece is ∆xi inches long, then the total length of the string is exactly n i=1 ∆xi . Summation notation can be used to represent physical quantities. Furthermore, notice that we can have an exact value even though we haven’t taken a limit. 10 / 14 Introduction to Integration
36. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check Suppose you are slicing an 11 inch long carrot REALLY thin from the greens end to the tip of the root. If each slice has a circular cross section f (x) = π(r(x))2 for each x between 0 and 11, and we make our cuts at x1, x2, x3, ... , xn, then a good approximation for the volume of the carrot is 1 n i=1 f (xi )xi 2 n i=1 [f (xi+1) − f (xi )]xi 3 n i=1 f (xi )[xi+1 − xi ] 11 / 14 Introduction to Integration
37. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check Suppose you are slicing an 11 inch long carrot REALLY thin from the greens end to the tip of the root. If each slice has a circular cross section f (x) = π(r(x))2 for each x between 0 and 11, and we make our cuts at x1, x2, x3, ... , xn, then a good approximation for the volume of the carrot is 1 n i=1 f (xi )xi 2 n i=1 [f (xi+1) − f (xi )]xi 3 n i=1 f (xi )[xi+1 − xi ] 11 / 14 Introduction to Integration
38. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #3 A table of values for the function f (x) is given below. x 0 2 4 6 8 10 12 f (x) −4 −2 1 6 7 8 10 Use the table to estimate the area between the graph of f (x) and the x-axis over the interval [0, 12]. 12 / 14 Introduction to Integration
39. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #3 A table of values for the function f (x) is given below. x 0 2 4 6 8 10 12 f (x) −4 −2 1 6 7 8 10 Use the table to estimate the area between the graph of f (x) and the x-axis over the interval [0, 12]. L6 = (−4 − 2 + 1 + 6 + 7 + 8)(2) = 32 R6 = (−2 + 1 + 6 + 7 + 8 + 10)(2) = 60 12 / 14 Introduction to Integration
40. How do we measure distance traveled? Summation Notation The Definite Integral Worksheet #3 A table of values for the function f (x) is given below. x 0 2 4 6 8 10 12 f (x) −4 −2 1 6 7 8 10 Use the table to estimate the area between the graph of f (x) and the x-axis over the interval [0, 12]. L6 = (−4 − 2 + 1 + 6 + 7 + 8)(2) = 32 R6 = (−2 + 1 + 6 + 7 + 8 + 10)(2) = 60 We can average the two estimates as well. L6 + R6 2 = 32 + 60 2 = 46 12 / 14 Introduction to Integration
41. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check You want to estimate the area underneath the graph of a positive function by using four rectangles of equal width. The rectangles that must give the best estimate of this area are those with height obtained from the: 1 Left endpoints 2 Midpoints 3 Right endpoints 4 Not enough information 13 / 14 Introduction to Integration
42. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check You want to estimate the area underneath the graph of a positive function by using four rectangles of equal width. The rectangles that must give the best estimate of this area are those with height obtained from the: 1 Left endpoints 2 Midpoints 3 Right endpoints 4 Not enough information 13 / 14 Introduction to Integration
43. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check Let f be a continuous function on the interval [a, b]. True or False: lim n→∞ n i=1 f (x∗ i )∆x. may lead to different limits if we choose the x∗ i to be the left- endpoints instead of midpoints. 14 / 14 Introduction to Integration
44. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check Let f be a continuous function on the interval [a, b]. True or False: lim n→∞ n i=1 f (x∗ i )∆x. may lead to different limits if we choose the x∗ i to be the left- endpoints instead of midpoints. 14 / 14 Introduction to Integration
45. How do we measure distance traveled? Summation Notation The Definite Integral Quick Check Let f be a continuous function on the interval [a, b]. True or False: lim n→∞ n i=1 f (x∗ i )∆x. may lead to different limits if we choose the x∗ i to be the left- endpoints instead of midpoints. When we are taking the limit as n goes to infinity, we can choose any sample points that we want. 14 / 14 Introduction to Integration