Part II: The Answers
406
Answers
601–700
2
3 5
3
5
3 0 5
2
1 2
0
2
1 2
0
0
0
e
e
dx
e
x
e
e
e
x
x
x
x
− −
(
)
=
−
+
(
)
= −
+
( )
( )
∫ ln
ln
( )
− − ( ) − ( ) +
= + − ( ) + ( ) − (
( )
( )
e
e
ln
ln
ln
ln
1 2
2
1 2
2
3
1
2
5
1 5 1
2
3
1
2
5 1
2 ) )
= − + ( ) −
=
+ ( )
6 1
4
1
2
5
2
13
4
1
8
3
ln
ln
662.
π
9
Recall the definition of volume: Let S be a solid that lies between x = a and x = b. If A(x)
is the cross-sectional area of S in the plane P x that goes through x and is perpendicular
to the x-axis, where A is a continuous function, then the volume of S is
V
A x dx
a
b
= ∫ ( )
Here’s the region that’s being rotated. Because the cross-sectional slice is a circle, you
have A(x) = π(y)
2
= π(f
(x))
2
.
Note that when y = 0, you have 0 = x
4
so that x = 0 is the lower limit of integration.
The integral to find the volume is
406
Answers
601–700
2
3 5
3
5
3 0 5
2
1 2
0
2
1 2
0
0
0
e
e
dx
e
x
e
e
e
x
x
x
x
− −
(
)
=
−
+
(
)
= −
+
( )
( )
∫ ln
ln
( )
− − ( ) − ( ) +
= + − ( ) + ( ) − (
( )
( )
e
e
ln
ln
ln
ln
1 2
2
1 2
2
3
1
2
5
1 5 1
2
3
1
2
5 1
2 ) )
= − + ( ) −
=
+ ( )
6 1
4
1
2
5
2
13
4
1
8
3
ln
ln
662.
π
9
Recall the definition of volume: Let S be a solid that lies between x = a and x = b. If A(x)
is the cross-sectional area of S in the plane P x that goes through x and is perpendicular
to the x-axis, where A is a continuous function, then the volume of S is
V
A x dx
a
b
= ∫ ( )
Here’s the region that’s being rotated. Because the cross-sectional slice is a circle, you
have A(x) = π(y)
2
= π(f
(x))
2
.
Note that when y = 0, you have 0 = x
4
so that x = 0 is the lower limit of integration.
The integral to find the volume is