Part II: The Answers
408
Answers
601–700
664.
2
15
π
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
= ∫ ( )
Because the cross-sectional slice is a circle in this case, you have A(x) = π(y)
2
= π(f
(x))
2
.
Here, the limits of integration correspond to the lines x = 3 and x = 5. Because y x
= 1 ,
the integral to find the volume is
π
π
π
π
π
π
1
1
1
5
1
3
2
15
3
5
2
2
3
5
1
3
5
3
5
x
dx
x dx
x
x
( ) =
= −
( )
= − ( )
= −
−
( )
=
∫
∫
−
−
665.
π ln 3
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
= ∫ ( )
Because the cross-sectional slice is a circle in this case, you have A(x) = π(y)
2
= π(f
(x))
2
.
Here, the limits of integration correspond to the lines x = 1 and x = 3. Because y
x
= 1 ,
the integral to find the volume is
π
π
π
π
π
1
1
3
1
3
2
1
3
1
3
1
3
x
dx
x
dx
x
=
=
=
−
=
∫
∫
ln
(ln
ln )
ln
666.
π
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
= ∫ ( )
408
Answers
601–700
664.
2
15
π
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
= ∫ ( )
Because the cross-sectional slice is a circle in this case, you have A(x) = π(y)
2
= π(f
(x))
2
.
Here, the limits of integration correspond to the lines x = 3 and x = 5. Because y x
= 1 ,
the integral to find the volume is
π
π
π
π
π
π
1
1
1
5
1
3
2
15
3
5
2
2
3
5
1
3
5
3
5
x
dx
x dx
x
x
( ) =
= −
( )
= − ( )
= −
−
( )
=
∫
∫
−
−
665.
π ln 3
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
= ∫ ( )
Because the cross-sectional slice is a circle in this case, you have A(x) = π(y)
2
= π(f
(x))
2
.
Here, the limits of integration correspond to the lines x = 1 and x = 3. Because y
x
= 1 ,
the integral to find the volume is
π
π
π
π
π
1
1
3
1
3
2
1
3
1
3
1
3
x
dx
x
dx
x
=
=
=
−
=
∫
∫
ln
(ln
ln )
ln
666.
π
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
= ∫ ( )