421
Answers
601–700
Answers and Explanations
where the outer radius, r out , is the distance of the expression farther away from the line
of rotation and the inner radius, r in , is the distance of the expression closer to the line
of rotation at a particular value of y.
Begin by setting the expressions equal to each other to find the points of intersection,
which correspond to the limits of integration:
y
y
2
4
2 2
=
= − ,
Because you’re rotating the region about the line x = 5, you have r out = 5 – y
2
and
r in = 5 – 4 = 1. Therefore, the integral to find the volume is
π
π
π
5
5 4
24 10
2
24 10
2
2
2
2
2
2
4
2
2
2
−
(
) − −
(
)

 

 
=
−
+
(
)
=
−
+
−
−
∫
∫
y
d y
y
y dy
y
y y dy
y
y
y
4
0
2
3
5
0
2
3
5
2 24
10
3
5
2 24 2
10 2
3
2
5
(
)
=
−
+






=
−
+


∫
π
π
( )
( )
( )
 



= 832
15
π
681.
5
2
1
2
2
2
−
−






e
e
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
= ∫ ( )
Here’s the region that’s being rotated: