92 Part I: The Questions
693. The region is bounded by the curves y = x
4
,
y = 16, and x = 0 and is rotated about the
x-axis.
694. The region is bounded by the curves
x = 5y
2
– y
3
and x = 0 and is rotated about
the x-axis.
695. The region is bounded by the curves y = x
2
and y = 4x – x
2
and is rotated about the line
x = 4.
696. The region is bounded by the curves
y = 1 + x + x
2
, x = 0, x = 1, and y = 0 and is
rotated about the y-axis.
697. The region is bounded by the curves
y = 4x – x
2
, x = 0, and y = 4 and is rotated
about the y-axis.
698. The region is bounded by the curves
y
x
=
−
9
, x = 0, and y = 0 and is rotated
about the x-axis.
Finding Volumes Using
Cylindrical Shells
688–711 Find the volume of the region bounded by
the given functions using cylindrical shells. Give an
exact answer. (Tip: Making a rough sketch of the
region that’s being rotated is often useful.)
688. The region is bounded by the curves y x
1 ,
y = 0, x = 1, and x = 3 and is rotated about
the y-axis.
689. The region is bounded by the curves y = x
2
,
y = 0, and x = 2 and is rotated about the
y-axis.
690. The region is bounded by the curves x y
= 3 ,
x = 0, and y = 1 and is rotated about the
x-axis.
691. The region is bounded by the curves y = x
2
,
y = 0, and x = 2 and is rotated about the line
x = –1.
692. The region is bounded by the curves y = 2x
and y = x
2
– 4x and is rotated about the
y-axis.