Part II: The Answers
410
Answers
601–700
668.
π
105
Recall the definition of volume: Let S be a solid that lies between y = c and y = d. If A(y)
is the cross-sectional area of S in the plane P y that goes through y and is perpendicular
to the y-axis, where A is a continuous function, then the volume of S is
V
A y dx
c
d
= ∫ ( )
Here’s the region that’s being rotated. Because the cross-sectional slice is a circle, you
have A(y) = π(x)
2
= π(g(y))
2
.
Begin by setting the expressions equal to each other and solving for y in order to find
the limits of integration:
0
0
1
0 1
2
3
2
=
−
=
−
=
y
y
y
y
y
(
)
,
Therefore, the integral to find the volume is
π
π
π
π
y
y dy
y
y
y dy
y
y
y
2
3
0
1
2
4
5
6
0
1
5
6
7
0
1
5
2
5
3
7
1
5
−
(
) =
−
+
(
)
=
−
+
=
∫
∫
− − +
=
1
3
1
7
105
6
7
π
410
Answers
601–700
668.
π
105
Recall the definition of volume: Let S be a solid that lies between y = c and y = d. If A(y)
is the cross-sectional area of S in the plane P y that goes through y and is perpendicular
to the y-axis, where A is a continuous function, then the volume of S is
V
A y dx
c
d
= ∫ ( )
Here’s the region that’s being rotated. Because the cross-sectional slice is a circle, you
have A(y) = π(x)
2
= π(g(y))
2
.
Begin by setting the expressions equal to each other and solving for y in order to find
the limits of integration:
0
0
1
0 1
2
3
2
=
−
=
−
=
y
y
y
y
y
(
)
,
Therefore, the integral to find the volume is
π
π
π
π
y
y dy
y
y
y dy
y
y
y
2
3
0
1
2
4
5
6
0
1
5
6
7
0
1
5
2
5
3
7
1
5
−
(
) =
−
+
(
)
=
−
+
=
∫
∫
− − +
=
1
3
1
7
105
6
7
π