AP Calculus AB



AP Calculus AB

Chapter 4: Numerical Integration

Worksheet #1

Write out the expansion for the indicated sum (you may use ellipses).

Use the programs on your calculator to find the value of the sum accurate to 3 decimal places.

1. Right Riemann Sum. Use n = 50 equal subdivisions. Write the sigma notation.

[pic] [2, 3]

2. Midpoint Riemann Sum. Use n = 40 equal subdivisions. Write the sigma notation.

[pic] [2, 3]

3. Trapezoidal Rule. Use 5 equal subdivisions.

[pic] [-2, 0]

4. Simpson’s Rule. Use 6 equal subdivisions.

[pic] [-2, 0]

5. Trapezoidal Rule. Use 8 equal subdivisions.

[pic] [1, 5]

6. Simpson’s Rule. Use 8 equal subdivisions.

[pic] [1, 5]

Identify the following for each expansion: [pic], and the type of sum.

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

AP Calculus AB

Chapter 4: Integration

Worksheet #2

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

Answers:

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

21.

AP Calculus AB

Chapter 4: U-Substitution

Worksheet #3

Find each antiderivative:

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

Answers:

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

AP Calculus AB

Chapter 4: Integration: Bounded Areas

Worksheet #4 (Area is positive, right?)

Find the area bounded by:

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

Answers:

1. 68

2. 256/5

3. 32/3

4. 8

5. 12

6. 48

7. 16/3

8. 2

9. 128/15

AP Calculus AB

Chapter 4: Integration

Worksheet #5

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

AP Calculus AB

Chapter 4: Final Review Worksheet

Worksheet #6

Evaluate each integral:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic]

6. Find [pic]if [pic] [pic]

7. Find the area of the region(s) bounded by the function: [pic] and the x-axis.

8. Which of the following statements is true of the function: [pic]

A) [pic] B) [pic]

C) [pic] D) [pic]

9. Evaluate: [pic]

10. Find the Average Value of the function: [pic] on [0, 2].

11. Find the area of the region bounded by[pic].

12. Write the formula for the each of the following area approximations, and then find the approximation to 3 decimal places. Use n = 4. Sketch a graph of the function and the rectangles for part "A" only. [pic]; on [3, 5].

A) Left Riemann Sum B) Trapezoidal Rule C) Simpson’s Rule

13. Explain the difference between an Upper Sum and a Right Sum. Show how they might be exactly the same, and then how/when they are different.

14. A) Find: [pic] B) Evaluate: [pic]

15. Sammy is having difficulty with two similar concepts in calculus. First he calculates: [pic]=0. Secondly, he finds by geometric analysis of the area bounded by the function and the x-axis he is 36. These two differing values of 0 and 36 are troublesome to him. Explain what his difficulty is, and how you would lead him to a better understanding of both concepts

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