Chapter 5 | Integration
523
5.1 EXERCISES 1. State whether the given sums are equal or unequal. a. ∑ i =1 10 i and ∑ k =1 10 k b. ∑ i =1 10 i and ∑ i =6 15 ( i −5) c. ∑ i =1 10 i ( i −1) and ∑ j =0 9 ⎛ ⎝ j +1 ⎞ ⎠ j d. ∑ i =1 10 i ( i −1) and ∑ k =1 10 ⎛ ⎝ k 2 − k ⎞ ⎠ In the following exercises, use the rules for sums of powers of integers to compute the sums. 2. ∑ i =5 10 i 3. ∑ i =5 10 i 2 Suppose that ∑ i =1 100 a i =15 and ∑ i =1 100 b i =−12. In the following exercises, compute the sums. 4. ∑ i =1 100 ⎛ ⎝ a i + b i ⎞ ⎠
20 ⎛
⎝ j 2 −10 j ⎞ ⎠
10. ∑
j =11
25 ⎡
⎣ (2 k ) 2 −100 k ⎤ ⎦
11. ∑ k =1
Let L n denote the left-endpoint sum using n subintervals and let R n denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. 12. L 4 for f ( x ) = 1 x −1 on [2, 3] 13. R 4 for g ( x ) =cos( πx ) on [0, 1]
on ⎡
⎤ ⎦
14. L 6 for f ( x ) = 1
⎣ 2, 5
x ( x −1)
on ⎡
⎤ ⎦
15. R 6 for f ( x ) = 1
⎣ 2, 5
x ( x −1)
1 x 2 +1 1 x 2 +1
on [−2, 2]
16. R 4 for
on [−2, 2]
17. L 4 for
2 −2 x +1 on [0, 2]
18. R 4 for x
19. L 8 for x 2 −2 x +1 on [0, 2] 20. Compute the left and right Riemann sums— L 4 and R 4 , respectively—for f ( x ) = (2− | x |) on [−2, 2]. Compute their average value and compare it with the area under the graph of f . 21. Compute the left and right Riemann sums— L 6 and R 6 , respectively—for f ( x ) = (3− |3− x |) on ⎡ ⎣ 0, 6 ⎤ ⎦ . Compute their average value and compare it with the area under the graph of f . 22. Compute the left and right Riemann sums— L 4 and R 4 , respectively—for f ( x ) = 4− x 2 on [−2, 2] and compare their values. 23. Compute the left and right Riemann sums— L 6 and R 6 , respectively—for f ( x ) = 9−( x −3) 2 on ⎡ ⎣ 0, 6 ⎤ ⎦ and compare their values. Express the following endpoint sums in sigma notation but do not evaluate them.
5. ∑ i =1 100 6. ∑ i =1 100 7. ∑ i =1 100
⎛ ⎝ a i − b i ⎞ ⎠
⎛ ⎝ 3 a i −4 b i ⎞ ⎠
⎛ ⎝ 5 a i +4 b i ⎞ ⎠
In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. 8. ∑ k =1 20 100 ⎛ ⎝ k 2 −5 k +1 ⎞ ⎠
50 ⎛
⎞ ⎠
9. ∑ j =1
⎝ j 2 −2 j
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