443
Answers
701–800
Answers and Explanations
Here’s the region being rotated about the line x = –1:
Begin by writing x = y
3
as x
1/3
= y. Then set the functions equal to each other to find
the points of intersection, which give you the limits of integration. Cube both sides
of the equation and factor to solve for x:
x
x
x x
x
x
x x
x
1 3
2
6
6
5
0
0
1
0 1
=
=
=
−
=
−
(
)
= ,
Notice that x
1/3
> x
2
on the interval [0, 1], so the shell height is (x
1/3
– x
2
). For a value
in the interval [0, 1], the distance from x to the line of rotation x = –1 is (x + 1), so the
shell radius is (x + 1). Therefore, the integral to find the volume of revolution using
cylindrical shells is
2
1
2
2 3
7
1
4
3
0
1
1 3
2
4 3
3
13
2
0
1
7 3
4
π
π
π
(
)
x
x
x dx
x
x
x
x dx
x
x
+
−
(
)
=
− +
−
(
)
=
−
+
∫
∫
4 4
1
3
2 3
7
1
4
3
4
1
3
25
21
4 3
3
0
1
x
x
−
)
(
=
− + −
(
)
=
π
π