Chapter 5 | Integration
545
−2 3
74.
83. ∫
(3− | x |) dx
In the following exercises, use averages of values at the left ( L ) and right ( R ) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. 84. {(0, 0), (2, 1), (4, 3), (5, 0), (6, 0), (8, 3)} over [0, 8]
{(0, 2), (1, 0), (3, 5), (5, 5), (6, 2), (8, 0)} over
85.
[0, 8]
{(−4, −4), (−2, 0), (0, −2), (3, 3), (4, 3)} over
86.
75.
[−4, 4]
{(−4, 0), (−2, 2), (0, 0), (1, 2), (3, 2), (4, 0)}
87.
over [−4, 4]
Suppose that ∫ 0 4
f ( x ) dx =5 and ∫ 0 2
f ( x ) dx =−3, and
4 g ( x ) dx =−1 and ∫ 0 2
∫
g ( x ) dx =2. In the following
0
exercises, compute the integrals.
4
88. ∫
⎛ ⎝ f ( x )+ g ( x ) ⎞
⎠ dx
In the following exercises, evaluate the integral using area formulas. 76. ∫ 0 3 (3− x ) dx 77. ∫ 2 3 (3− x ) dx 78. ∫ −3 3 (3− | x |) dx 79. ∫ 0 6 (3− | x −3|) dx 80. ∫ −2 2 4− x 2 dx 81. ∫ 1 5 4−( x −3) 2 dx 82. ∫ 0 12 36−( x −6) 2 dx
0
4
89. ∫
⎛ ⎝ f ( x )+ g ( x ) ⎞
⎠ dx
2
2
90. ∫
⎛ ⎝ f ( x )− g ( x ) ⎞
⎠ dx
0
4
91. ∫
⎛ ⎝ f ( x )− g ( x ) ⎞
⎠ dx
2
2
92. ∫
⎛ ⎝ 3 f ( x )−4 g ( x ) ⎞
⎠ dx
0
4
93. ∫
⎛ ⎝ 4 f ( x )−3 g ( x ) ⎞
⎠ dx
2
In the following exercises,
use the identity A f ( x ) dx to compute the
− A 0
− A A
∫
f ( x ) dx = ∫
f ( x ) dx + ∫
0
integrals. 94. ⌠ ⌡ − π π
sin t 1+ t 2
dt ( Hint : sin(− t ) =−sin( t ))
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