834
Answer Key
⌠ ⌡ 1 4
4 x 2
169 . T 10 =3.058,
dx =3
171 . F ( x ) = x 3 173 . F ( x ) = − t 5 5 3 + 175 . F ( x ) = x 177 . F ( x ) = x 3 100 100 ,
3 x 2
x , F (3)− F (−2) = − 35 6
2 −5
3 3 −36
+ 13 t
t , F (3)− F (2) = 62 15
F (1)− F (0) = 1 100
1 x , F (4)− F ⎛
⎞ ⎠ = 1125 64
⎝ 1 4
3 +
179 . F ( x ) = x , F (4)− F (1) =1 181 . F ( x ) = 4 3 183 . F ( x ) =−cos x , F ⎛ ⎝ π 2 ⎞
t 3/4 , F (16)− F (1) = 28 3
⎠ − F (0) =1
185 . F ( x ) = sec x , F ⎛ ⎝ π 4 ⎞
⎠ − F (0) = 2−1
187 . F ( x ) =−cot( x ), F ⎛ ⎝ π 2 ⎞
⎛ ⎝ π 4
⎞ ⎠ =1
⎠ − F
189 . F ( x ) = − 1 x + 1 2 x 2
, F (−1)− F (−2) = 7 8
191 . F ( x ) = e x − e 193 . F ( x ) =0 195 . ∫ −2 −1 ⎛
3 ⎛
4 ⎛ ⎝ t 2 −2 t −3 ⎞
⎝ t 2 −2 t −3 ⎞
⎝ t 2 −2 t −3 ⎞
⎠ dt − ∫
⎠ dt + ∫
⎠ dt = 46 3
−1
3
⌡ − π /2 0
197 . − ⌠
π /2
sin tdt + ∫
sin tdt =2
0
199 . a. The average is 11.21×10 9 since cos ⎛ ⎝ πt 6 ⎞
⎠ has period 12 and integral 0 over any period. Consumption is equal to
the average when cos ⎛ ⎝ πt 6 ⎞
⎠ =0, when t =3, and when t =9. b. Total consumption is the average rate times duration: 11.21 × 12 × 10 9 =1.35×10 11 c. 10 9 ⎛ ⎝ ⎠ =11.84 x 10 9 201 . If f is not constant, then its average is strictly smaller than the maximum and larger than the minimum, which are attained over ⎡ ⎣ a , b ⎤ ⎦ by the extreme value theorem. 203 . a. d 2 θ = ( a cos θ + c ) 2 + b 2 sin 2 θ = a 2 + c 2 cos 2 θ +2 ac cos θ = ( a + c cos θ ) 2 ; b. ⎜ 11.21− 1 6 ⌠ ⌡ 3 9 cos ⎛ ⎝ πt 6 ⎞ ⎠ dt ⎞ ⎠ ⎟ =10 9 ⎛ ⎝ 11.21+ 2 π ⎞
2 π
d –
= 1 2 π ∫
( a +2 c cos θ ) dθ = a
0
⌠ ⌡ ⎮⎮ 0 2 π
205 . Mean gravitational force = GmM 2
1
dθ .
⎛ ⎝ a +2 a 2 − b 2 cos θ ⎞ ⎠
2
⎛ ⎝ x − 1 x
⎞ ⎠ dx = ∫ x 1/2 dx − ∫ x −1/2 dx = 2 3
207 . ∫ 209 . ⌠ ⌡ 211 . ⌠
x 3/2 + C
1/2 + C
x 3/2 −2 x 1/2 + C
1 −2 x
2 = 2 3
dx 2 x = 1 2 ln|
x | + C
⌡ 0 π
π cos xdx =−cos x | 0
sin xdx − ∫
π −(sin x ) |
0 π = ⎛
⎝ −(−1)+1 ⎞
⎠ −(0−0) =2
0
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