Part II: The Answers
466
Answers
701–800
742.
7
2
10
2
−
(
)
Using the average value formula with a = 0, b = 8, and f x
x
( ) =
+
14
2
gives you
f
x
dx
x
dx
avg = −
+
=
+
∫
∫
1
8 0
14
2
1
8
14
2
0
8
0
8
Next, use the substitution u = x + 2 so that du = dx. You can find the new limits of
integration by noting that if x = 8, then u = 8 + 2 = 10, and if x = 0, then u = 0 + 2 = 2.
With these new values, you find that
1
8
14
7
4
2
7
2
10
2
1 2
2
10
1 2
2
10
u du
u
−
∫
= ( )
=
−
(
)
743.
d = 1
Using the average value formula with a = 0, b = d, and f
 
(x) = 2 + 4x – 3x
2
gives you
f
d
x
x dx
d
x
x x
d
d d d
d
d
avg = −
+
−
(
)
=
+
−
(
)
=
+
−
(
)
= +
∫
1
0
2 4
3
1 2
2
1 2
2
2 2
2
0
2
3
0
2
3
d d d
−
2
Next, set this expression equal to 3 and solve for d:
2 2
3
0
2 1
0
1
1
1
2
2
+ −
=
=
− +
= −
−
=
d d
d
d
d
d
d
(
)(
)
744.
d = 4
Using the average value formula with a = 0, b = d, and f
 
(x) = 3 + 6x – 9x
2
gives you
f
d
x
x dx
d
x
x
x
d
d
d
d
d
d
avg = −
+
−
(
)
=
+
−
(
)
=
+
−
(
)
=
∫
1
0
3 6
9
1 3
3
3
1 3
3
3
3
2
0
2
3
0
2
3
+ +
−
3
3
2
d
d