546
Chapter 5 | Integration
− π π
π /2
t 1+cos t
sin tdt ≥ π
Hint : sin t ≥ 2 t
95. ∫
108. Show that ∫ 0
dt
π over
4 . (
⎡ ⎣ 0, π 2
⎤ ⎦ )
In the following exercises, find the net signed area between f ⎛ ⎝ x ⎞ ⎠ and the x-axis.
− π /4 π /4
109. Show that ∫
cos tdt ≥ π 2/4.
3 (2− x ) dx ( Hint: Look at the graph of f .)
96. ∫
1
In the following exercises, find the average value f ave of f between a and b , and find a point c , where f ( c ) = f ave .
4 ( x −3) 3 dx ( Hint: Look at the graph of f .)
97. ∫
2
110. f ( x ) = x 2 , a =−1, b =1 111. f ( x ) = x 5 , a =−1, b =1 112. f ( x ) = 4− x 2 , a =0, b =2 113. f ( x ) = (3− | x |), a =−3, b =3 114. f ( x ) = sin x , a =0, b =2 π 115. f ( x ) =cos x , a =0, b =2 π
In the following exercises,
given that
1
1
1
∫
2 , ∫
∫
x 3 dx = 1
x 2 dx = 1
xdx = 1
and
3 ,
4 ,
0
0
0
compute the integrals.
1 ⎛ ⎝ 1+ x + x 2 + x 3 ⎞
98. ∫
⎠ dx
0
1 ⎛ ⎝ 1− x + x 2 − x 3 ⎞
99. ∫
⎠ dx
0
1 (1− x ) 2 dx
In the following exercises, approximate the average value using Riemann sums L 100 and R 100 . How does your answer compare with the exact given answer? 116. [T] y = ln( x ) over the interval [1, 4]; the exact solution is ln(256) 3 −1. 117. [T] y = e x /2 over the interval [0, 1]; the exact solution is 2( e −1).
100. ∫
0
1 (1−2 x ) 3 dx
101. ∫
0
1 ⎛
x 2 ⎞
102. ⌠
⎝ 6 x − 4 3
⎠ dx
⌡ 0
1 ⎛
⎝ 7−5 x 3 ⎞
103. ∫
⎠ dx
118. [T] y = tan x over the interval ⎡
⎤ ⎦ ; the exact
⎣ 0, π 4
0
solution is 2ln(2) π . 119. [T] y = x +1 4− x 2 exact solution is π 6 .
In the following exercises, use the comparison theorem . 104. Show that ∫ 0 3 ⎛ ⎝ x 2 −6 x +9 ⎞ ⎠ dx ≥0. 105. Show that ∫ −2 3 ( x −3)( x +2) dx ≤0. 106. Show that ∫ 0 1 1+ x 3 dx ≤ ∫ 0 1 1+ x 2 dx . 107. Show that ∫ 1 2 1+ xdx ≤ ∫ 1 2 1+ x 2 dx .
over the interval [−1, 1]; the
In the following exercises, compute the average value using the left Riemann sums L N for N = 1, 10, 100. Howdoes the accuracy compare with the given exact value? 120. [T] y = x 2 −4 over the interval [0, 2]; the exact solution is − 8 3 .
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