1. Splash Screen

2. Which of the following represents the derivative of h(x) = (4x2 – 3x + 5)(2x – 7)?
A. h'(x) = 8x2 + 10x + 4
B. h'(x) = 24x2 – 68x + 31
C. h'(x) = 8x3 – 34x2 + 57x – 41
D. h'(x) = 32x3 – 36x2 + 53x – 29
5–Minute Check 4

3. Which of the following represents the derivative of h(x) = (4x2 – 3x + 5)(2x – 7)?
A. h'(x) = 8x2 + 10x + 4
B. h'(x) = 24x2 – 68x + 31
C. h'(x) = 8x3 – 34x2 + 57x – 41
D. h'(x) = 32x3 – 36x2 + 53x – 29
5–Minute Check 4

4. Approximate the area under a curve using rectangles.
You computed limits algebraically using the properties of limits. (Lesson 12-2)
Approximate the area under a curve using rectangles.
Approximate the area under a curve using definite integrals and integration.
Then/Now

5. regular partition definite integral lower limit upper limit
right Riemann sum
integration
Vocabulary

6. Area Under a Curve Using Rectangles
Approximate the area between the curve f(x) = –x2 + 18x and the x-axis on the interval [0, 18] using 6, 9, and 18 rectangles. Use the right endpoint of each rectangle to determine the height.
Example 1

7. Area using 6 rectangles total area = 945
Area Under a Curve Using Rectangles
Area using 6 rectangles
total area = 945
Example 1

8. Area using 9 rectangles total area = 960
Area Under a Curve Using Rectangles
Area using 9 rectangles
total area = 960
Example 1

9. Area Under a Curve Using Rectangles
Area using 18 rectangles
Example 1

10. Area Under a Curve Using Rectangles
total area = 969
Example 1

11. A. 2 rectangles = 125 units2; 5 rectangles = 160 units2
Approximate the area between the curve f(x) = –x2 + 10x and the x-axis on the interval [0, 10] using 2 and 5 rectangles. Use the right endpoint of each rectangle to determine the height.
A. 2 rectangles = 125 units2; 5 rectangles = 160 units2
B. 2 rectangles = 125 units2; 5 rectangles = 128 units2
C. 2 rectangles = 25 units2; 5 rectangles = 128 units2
D. 2 rectangles = 16 units2; 5 rectangles = 160 units2
Example 1

12. Area Under a Curve Using Left and Right Endpoints
Approximate the area between the curve f(x) = x2 + 1 and the x-axis on the interval [0, 4] by first using the right endpoints and then by using the left endpoints of the rectangles. Use rectangles with a width of 1. Then find the average for both approximations.
Use the figures for reference. Using right endpoints for the height of each rectangle produces four rectangles with a width of 1 unit. Using left endpoints for the height of each rectangle produces four rectangles with a width of 1 unit.
Example 2

13. Area using right endpoints
Area Under a Curve Using Left and Right Endpoints
Area using right endpoints
total area = 34 units2
Example 2

14. Area using left endpoints
Area Under a Curve Using Left and Right Endpoints
Area using left endpoints
total area = 18 units2
Example 2

15. Area Under a Curve Using Left and Right Endpoints
Answer: The area using the right and left endpoints is 34 and 18 square units, respectively. We now have lower and upper estimates for the area of the region, 18 < area < 34. Averaging the two areas would give a better approximation of 26 square units.
Example 2

16. Approximate the area between the curve f(x) = 2x2 + 2 and the x-axis on the interval [0, 3] by first using the right endpoints and then by using the left endpoints of the rectangles. Use rectangles with a width of 1. Then find the average for both approximations.
A. right endpoint = 16 units2; left endpoint = 68 units2; average = 42 units2
B. right endpoint = 68 units2; left endpoint = 16 units2; average = 42 units2
C. right endpoint = 16 units2 left endpoint = 34 units2; average = 25 units2
D. right endpoint = 34 units2 left endpoint = 16 units2; average = 25 units2
Example 2

17. Key Concept 3

18. Area Under a Curve Using Integration
Use limits to find the area of the region between the graph of y = x2 + 1 and the x-axis on the interval [0, 4], or
Example 3

19. First, find x and xi. Formula for x b = 4 and a = 0 Formula for xi
Area Under a Curve Using Integration
First, find x and xi.
Formula for x
b = 4 and a = 0
Formula for xi
a = 0 and
Example 3

20. Calculate the definite integral that gives the area.
Area Under a Curve Using Integration
Calculate the definite integral that gives the area.
Definition of a definite integral
f(xi) = (xi)2 + 1
Example 3

21. Factor. Simplify. Apply summations.
Area Under a Curve Using Integration
Factor.
Simplify.
Apply summations.
Example 3

22. Factor constants. Summation formulas Distribute
Area Under a Curve Using Integration
Factor constants.
Summation formulas
Distribute
Example 3

23. Factor and perform division.
Area Under a Curve Using Integration
Simplify.
Factor and perform division.
Limit Theorems
Example 3

24. Limits Simplify. Answer: Area Under a Curve Using Integration
Example 3

25. Use limits to find the area of the region between the graph of y = 2x2 and the x-axis on the interval [0, 3], or
A. 9 units2
B. 18 units2
C. 27 units2
D. about units2
Example 3

26. Area Under a Curve Using Integration
Use limits to find the area of the region between the graph of y = x3 + 1 and the x-axis on the interval [2, 4], or
Example 4

27. First, find x and xi. Formula for x b = 4 and a = 2 Formula for xi
Area Under a Curve Using Integration
First, find x and xi.
Formula for x
b = 4 and a = 2
Formula for xi
a = 2 and
Example 4

28. Calculate the definite integral that gives the area.
Area Under a Curve Using Integration
Calculate the definite integral that gives the area.
Definition of a definite integral
f(xi) = (xi)3 + 1
Example 4

29. Factor. Expand. Simplify. Area Under a Curve Using Integration
Example 4

30. Apply summations. Factor constants.
Area Under a Curve Using Integration
Apply summations.
Factor constants.
Example 4

31. Area Under a Curve Using Integration
Summation formulas
Example 4

32. Area Under a Curve Using Integration
Distribute
Example 4

33. Factor and perform division.
Area Under a Curve Using Integration
Simplify.
Factor and perform division.
Example 4

34. Limit theorems. Simplify. Answer: 62 units2
Area Under a Curve Using Integration
Limit theorems.
Simplify.
Answer: 62 units2
Example 4

35. Use limits to approximate the area of the region between the graph of y = 2x3 + 3 and the x-axis on the interval [1, 3], or
A. 50 units2
B. 64 units2
C. 46 units2
D. 18 units2
Example 4

36. Area Under a Curve
BUSINESS A clothing manufacturer produces 2000 pairs of pants per day. The cost for increasing the number of pairs per day from to 5000 can be found by What is the increase in cost?
Example 5

37. First, find x and xi. Formula for x b = 5000 and a = 2000
Area Under a Curve
First, find x and xi.
Formula for x
b = 5000 and a = 2000
Formula for xi
a = 2000 and
Example 5

38. Calculate the definite integral that gives the area.
Area Under a Curve
Calculate the definite integral that gives the area.
Definition of a definite integral
f(xi) = 20 – 0.004xi
Example 5

39. Distributive property and simplify.
Area Under a Curve
Distributive property and simplify.
Example 5

40. Apply summations. Factor. Summation formulas. Area Under a Curve
Example 5

41. Factor and perform division.
Area Under a Curve
Distribute
Simplify.
Factor and perform division.
Example 5

42. Limit Theorems Simplify.
Area Under a Curve
Limit Theorems
Simplify.
Answer: The cost for increasing the number of pairs of pants produced per day from 2000 to 5000 pairs is $18,000.
Example 5

43. PAVING Millie is putting in a brick patio
PAVING Millie is putting in a brick patio. The paver charges $105 per square foot. If the area of Millie’s patio can be found by , to the nearest dollar, how much will the paver charge for installing the patio if x is given in feet?
A. $2,017.50
B. $3150
C. $3500
D. $4,120.10
Example 5

44. regular partition definite integral lower limit upper limit
right Riemann sum
integration
Vocabulary