Section 7.2: Volume
Let S be a solid and suppose that the area of the cross-section of S in the plane Px
perpendicular to the x-axis passing through x is A(x) for a ≤ x ≤ b.
Consider slicing the solid into cylindrical slabs with base area A(x) and width dx. The
volume of these slabs is dV = A(x)dx. Summing the volumes of these slabs for a ≤ x ≤ b,
we obtain the volume of S.
Definition: Let S be a solid with cross-sectional area A(x) perpendicular to the x-axis at
each point x ∈ [a, b]. The volume of S is
Z b
A(x)dx.
V =
a
Note: This method for computing the volume of a solid is known as the Method of Slicing.
Example: Find the volume of the solid whose base is the region bounded by y = ex , y = 0,
x = 1, and x = 3 where the cross-sections perpendicular to the x-axis are semicircles.
The volume of the slab is
1
π
dV = π[r(x)]2 dx =
2
2
ex
2
2
dx =
Then the volume of the solid is
V
Z
π 3 2x
=
e dx
8 1
π 2x 3
=
e 16
1
π 6
=
(e − e2 ).
16
π 2x
e dx.
8
Example: Find the volume of the solid whose base is the region bounded by y = 1 − x2 and
the x-axis where cross-sections perpendicular to the y-axis are squares.
The volume of the slab is
dV = [L(y)]2 dy = (2x)2 dy = 4x2 dy = (4 − 4y)dy.
Then the volume of the solid is
Z
V
1
(4 − 4y)dy
1
= (4y − 2y 2 )0
= 2.
=
0
Example: Find the volume of the solid whose base is the triangular region with vertices (0, 0),
(2, 0), and (0, 2) where cross-sections perpendicular to the x-axis are equilateral triangles.
The base of the solid is the region bounded by the lines y = 0, x = 0, and y = 2 − x. The
volume of the slab is
√
√
√
3
3
3
dV =
[L(x)]2 dx =
(2 − x)2 dx =
(4 − 4x + x2 )dx.
4
4
4
Then the volume of the solid is
V
=
=
=
=
√ Z 2
3
(4 − 4x + x2 )dx
4 0
√ 2
3
1 3 2
4x − 2x + x 4
3
0
√ 3 8
4
3
√
2 3
.
3
Definition: A solid of revolution is a solid S obtained by revolving a region R in the plane
about a line L called the axis of revolution.
Let f be a continuous function such that f (x) ≥ 0 on [a, b]. Suppose we want to find the
volume of the solid S obtained by revolving the region R under the curve y = f (x) on [a, b]
about the x-axis.
Consider a vertical strip of width dx. Revolving this strip about the x-axis, we obtain a disk
with volume
dV = π[r(x)]2 dx = π[f (x)]2 dx.
Summing the volumes of these disks, we obtain the volume of S.
Theorem: (The Disk Method)
Suppose R is the region under the curve y = f (x) on [a, b]. If R is revolved about the x-axis,
the volume of the resulting solid is
Z b
V =π
[f (x)]2 dx.
a
Example: Find the volume of the solid obtained by revolving the region bounded by y =
y = 0, and x = 4 about the x-axis.
√
x,
Using the Disk Method,
4
Z
V
= π
√
( x)2 dx
Z0 4
xdx
= π
0
π 2 4
x
2 0
= 8π.
=
Example: Find the volume of the solid obtained by revolving the region bounded by y = 1−x,
y = 0, and x = 0 about the x-axis.
Using the Disk Method,
Z
V
1
= π
(1 − x)2 dx
Z0 1
(1 − 2x + x2 )dx
1
1 3 2
= π x−x + x 3
0
π
=
.
3
= π
0
Example: Find the volume of the solid obtained by revolving the region bounded by y = ln x,
x = 0, y = 0, and y = 2 about the y-axis.
Using the Disk Method,
Z
V
2
(ey )2 dy
= π
0
Z
2
e2y dy
0
π 2y 2
=
e 2
0
π 4
=
(e − 1).
2
= π
Example: Find the volume of the solid obtained by revolving the region bounded by y =
y = 0, and x = 4 about the line x = 4.
√
x,
Using the Disk Method
2
Z
V
(4 − y 2 )2 dy
= π
Z0 2
(16 − 8y 2 + y 4 )dy
2
8 3 1 5 = π 16y − y + y 3
5
0
256π
=
.
15
= π
0
Example: Find the volume of the solid obtained by revolving the region bounded by y = x2
and y = 1 about the line y = 1.
Using the Disk Method,
Z
V
1
= π
(1 − x2 )2 dx
−1
Z
1
= 2π
(1 − x2 )2 dx
Z0 1
(1 − 2x2 + x4 )dx
0
1
2 3 1 5 = 2π x − x + x 3
5
0
16π
=
.
15
= 2π
Let f and g be continuous functions such that f (x) ≥ g(x) ≥ 0 on [a, b]. Suppose we want
to find the volume of the solid S obtained by revolving the region R between the curves
y = f (x) and y = g(x) on [a, b] about the x-axis. Revolving a vertical strip of width dx
about the x-axis, we obtain a washer with volume
dV = π [f (x)]2 − [g(x)]2 dx.
Summing the volume of these washers, we obtain the volume of S.
Theorem: (The Washer Method)
Suppose R is the region between the curves y = f (x) and y = g(x) on [a, b]. If R is revolved
about the x-axis, the volume of the resulting solid is
Z b
[f (x)]2 − [g(x)]2 dx.
V =π
a
Example: Find the volume of the solid obtained by revolving the region bounded by the
curves y = x2 and y = 2x about the x-axis.
Using the Washer Method,
Z
V
2
= π
(2x)2 − (x2 )2 dx
Z0 2
(4x2 − x4 )dx
0
2
4 3 1 5 = π
x − x 3
5
0
64π
=
.
15
= π
Example:√Find the volume of the solid obtained by revolving the region bounded by y = x2
and y = x about the x-axis.
Using the Washer Method,
Z
V
1
= π
Z0 1
√ 2
( x) − (x2 )2 dx
(x − x4 )dx
0
1
1 2 1 5 = π
x − x 2
5
0
3π
=
.
10
= π
Example: Find the volume of the solid obtained by revolving the region bounded by the
curves y = x2 and y = x about the line y = 2.
Using the Washer Method,
Z
V
1
(1 − x)2 − (1 − x2 )2 dx
= π
Z0 1
(x4 − 5x2 + 4x)dx
1
1 5 5 3
2 = π
x − x + 2x 5
3
0
8π
=
.
15
= π
0
Example: Find
√ the volume of the solid obtained by revolving the region bounded by the
curves y = x, y = 0, and x = 1 about the line x = 2.
Using the Washer Method,
Z
V
1
= π
Z0 1
(2 − y 2 )2 − (2 − 1)2 dy
(3 − 4y 2 + y 4 )dy
0
1
4 3 1 5 = π 3y − y + y 3
5
0
28π
=
.
15
= π