544
Chapter 5 | Integration
5.2 EXERCISES In the following exercises, express the limits as integrals.
71.
→∞ ∑ i =1 n →∞ ∑ i =1 →∞ ∑ i =1 n →∞ ∑ i =1 n
⎛ ⎝ x i *
⎞ ⎠ Δ x over [1, 3]
lim n
60.
n ⎛
⎞ ⎠ Δ x over [0, 2]
⎛ ⎝ x i *
⎞ ⎠ 2 −3
⎛ ⎝ x i *
⎞ ⎠ 3
lim n
⎝ 5
61.
sin 2 ⎛
⎞ ⎠ Δ x over [0, 1]
lim n
⎝ 2 πx i *
62.
cos 2 ⎛
⎞ ⎠ Δ x over [0, 1]
72.
lim n
⎝ 2 πx i *
63.
In the following exercises, given L n or R n as indicated, express their limits as n →∞ as definite integrals, identifying the correct intervals. 64. L n = 1 n ∑ i =1 n i −1 n 65. R n = 1 n ∑ i =1 n i n 66. L n = 2 n ∑ i =1 n ⎛ ⎝ 1+2 i −1 n ⎞ ⎠
73.
67. R n = 3 n ∑ i =1 n ⎛
⎞ ⎠
⎝ 3+3 i n
n ∑ i =1 n
⎛ ⎝ 2 π i −1 n
⎞ ⎠
68. L n = 2 π
2 π i −1
n cos
⎛ ⎝ ⎛
2 ⎞ ⎠
69. R n = 1 n ∑ i =1 n ⎛
⎞ ⎠ log
⎞ ⎠
⎝ 1+ i n
⎝ 1+ i n
In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x -axis. 70.
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