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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