Part II: The Answers
418
Answers
601–700
Here’s the region being rotated:
In this case, the cross-sectional slice has the shape of a washer. When the cross-sectional
slice is a washer, you can find A(x) by using
A
r
r
=
= ( )
( )
π
π
π
π
(
)
(
)
outer radius
inner radius
out
i n
2
2
2
2
−
−
where the outer radius, r out , is the distance of the function farther away from the line of
rotation and the inner radius, r in , is the distance of the function closer to the line of
rotation at a particular value of x.
Begin by finding the point of intersection by setting the functions equal to each other
and solving for x:
x
x
2 3
1
1
=
=
The limits of integration are therefore x = 0 (given) and x = 1. Note that r out = 2 – x
2/3
and
r in = 2 – 1 = 1 because you’re rotating the region about the line y = 2. Therefore, the
integral to find the volume is
π
π
π
2
2 1
3 4
3
12
5
2 3
2
2
0
1
2 3
4 3
0
1
5 3
−
(
) − −

 

 
=
−
+
(
)
=
−
+
∫
∫
x
d x
x
x
dx
x
x
(
)
3 3
7
3 12
5
3
7
36
35
7 3
0
1
x






=
−
+
(
)
=
π
π