find the volume of the solid enclosed by the curves y=x and y=x^2 when it is rotated about the line y=-1
Answers
Answer:
The disk method is used when we rotate a single curve
y
=
f
(
x
)
around the
x
−
(or
y
−
) axis.
Suppose that
y
=
f
(
x
)
is a continuous non-negative function on the interval
[
a
,
b
]
.The volume of the solid formed by revolving the region bounded by the curve
y
=
f
(
x
)
and the
x
−
axis between
x
=
a
and
x
=
b
about the
x
−
axis is given by
V
=
π
b
∫
a
[
f
(
x
)
]
2
d
x
.
The cross section perpendicular to the axis of revolution has the form of a disk of radius
R
=
f
(
x
)
.
Similarly, we can find the volume of the solid when the region is bounded by the curve
x
=
f
(
y
)
and the
y
−
axis between
y
=
c
and
y
=
d
,
and is rotated about the
y
−
axis.The resulting formula is
V
=
π
d
∫
c
[
f
(
y
)
]
2
d
y
.
The Washer Method
We can extend the disk method to find the volume of a hollow solid of revolution.
Assuming that the functions
f
(
x
)
and
g
(
x
)
are continuous and non-negative on the interval
[
a
,
b
]
and
g
(
x
)
≤
f
(
x
)
,
consider a region that is bounded by two curves
y
=
f
(
x
)
and
y
=
g
(
x
)
,
between
x
=
a
and
x
=
b
.
The volume of the solid formed by revolving the region about the
x
−
axis is
V
=
π
b
∫
a
(
[
f
(
x
)
]
2
−
[
g
(
x
)
]
2
)
d
x
.
At a point
x
on the
x
−
axis, a perpendicular cross section of the solid is washer-shape with the inner radius
r
=
g
(
x
)
and the outer radius
R
=
f
(
x
)
.
The volume of the solid generated by revolving about the
y
−
axis a region between the curves
x
=
f
(
y
)
and
x
=
g
(
y
)
,
where
g
(
y
)
≤
f
(
y
)
and
c
≤
y
≤
d
is given by the formula
V
=
π
d
∫
c
(
[
f
(
y
)
]
2
−
[
g
(
y
)
]
2
)
d
y
.If a bounding curve is defined in parametric form by the equations
x
=
x
(
t
)
,
y
=
y
(
t
)
,
where the parameter
t
varies from
α
to
β
,
then the volume of the solid generated by revolving the curve about the
x
−
axis is given by
V
x
=
π
β
∫
α
y
2
(
t
)
d
x
d
t
d
t
.
Respectively, when the curve is rotated about the
y
−
axis, the volume of the solid of revolution is equal
V
y
=
π
β
∫
α
x
2
(
t
)
d
y
d
t
d
t
.