Part II: The Answers
416
Answers
601–700
675.
108
5
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.
Begin by finding the points of intersection in order to find the limits of integration. Solve
the second equation for y, set the expressions equal to each other, and solve for x:
3
3 2
0 3
0
3
0 3
2
2
− = +
−
=
−
=
−
(
)
=
x
x x
x x
x
x
x
,
To find out which function is greater on the interval (0, 3) without graphing (and is therefore farther away from the line of rotation), take a point in this interval and substitute
that value into each function. So if x = 1, you have y = 3 + 2 – 1
2
= 4 and y = 3 – 1 = 2.
Therefore, r out = 3 + 2x – x
2
and r in = 3 – x, so the integral becomes
π
π
3 2
3
4
2
12
9
9 6
2
2
2
0
3
4
3
2
2
+
−
(
) − −
(
)
=
−
−
+
+
(
) − − +
(
∫
x x
x
dx
x
x
x
x
x x ) )
=
−
−
+
(
)
=
−
−
+
=
∫
∫
0
3
4
3
2
0
3
5
4
3
2
0
3
4
3
18
5
9
dx
x
x
x
x dx
x
x
x
x
π
π
π π
π
3
5
3 3 9 3
108
5
5
4
3
2
− − +
=
( )
676.
3
10
π
Recall the definition of volume: Let S be a solid that lies between y = c and y = d. If A(y)
is the cross-sectional area of S in the plane P y that goes through y and is perpendicular
to the y-axis, where A is a continuous function, then the volume of S is
V
A y dx
c
d
= ∫ ( )
416
Answers
601–700
675.
108
5
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.
Begin by finding the points of intersection in order to find the limits of integration. Solve
the second equation for y, set the expressions equal to each other, and solve for x:
3
3 2
0 3
0
3
0 3
2
2
− = +
−
=
−
=
−
(
)
=
x
x x
x x
x
x
x
,
To find out which function is greater on the interval (0, 3) without graphing (and is therefore farther away from the line of rotation), take a point in this interval and substitute
that value into each function. So if x = 1, you have y = 3 + 2 – 1
2
= 4 and y = 3 – 1 = 2.
Therefore, r out = 3 + 2x – x
2
and r in = 3 – x, so the integral becomes
π
π
3 2
3
4
2
12
9
9 6
2
2
2
0
3
4
3
2
2
+
−
(
) − −
(
)
=
−
−
+
+
(
) − − +
(
∫
x x
x
dx
x
x
x
x
x x ) )
=
−
−
+
(
)
=
−
−
+
=
∫
∫
0
3
4
3
2
0
3
5
4
3
2
0
3
4
3
18
5
9
dx
x
x
x
x dx
x
x
x
x
π
π
π π
π
3
5
3 3 9 3
108
5
5
4
3
2
− − +
=
( )
676.
3
10
π
Recall the definition of volume: Let S be a solid that lies between y = c and y = d. If A(y)
is the cross-sectional area of S in the plane P y that goes through y and is perpendicular
to the y-axis, where A is a continuous function, then the volume of S is
V
A y dx
c
d
= ∫ ( )