411
Answers
601–700
Answers and Explanations
669.
8π
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
= ∫ ( )
Because the cross-sectional slice is a circle in this case, you have A(x) = π(y)
2
= π(f
 
(x))
2
.
To get the lower limit of integration, find where the function y
x
=
−1 intersects the
line y = 0. Set the equations equal to each other and solve for x:
0
1
0
1
1
=
−
= −
=
x
x
x
Therefore, x = 1 and x = 5 are the limits of integration. Because y
x
=
−1, the integral
becomes
π
π
π
π
x
dx
x
dx
x
x
−
(
) =
−
=
−






=
−





 −
−
∫
∫
1
1
2
5
2
5
1
2
1
1
5
2
1
5
2
1
5
2
2
(
)
 











= 8π
670.
448 2
15
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
= ∫ ( )
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.