Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
Let λ be a horizontal or vertical axis of rotation in the same plane as a plane region A, no interior point of which is on λ. A solid of revolution is formed when A is rotated about λ.
The volume of the solid of revolution can be found with definite integrals formed by slicing the solid either into disks or cylindrical shells. Table 5.2.1 lists the types of definite integrals that arise in the calculation of the volume of a solid of revolution.
π ∫abρ2x ⅆx
π ∫cdρ2y ⅆy
ρ is the (varying) radius of rotation, the distance from the axis of rotation λ, to the boundary of the rotated region A.
(Disks with holes)
π ∫abR2x−r2x ⅆx
π ∫cdR2y−r2y ⅆy
R is the radius of rotation of the outer boundary of A, whereas r is the radius of rotation of the inner boundary.
2 π ∫abρx Lx ⅆx
2 π ∫cdρy Ly ⅆy
ρ is the (varying) radius of the shells.
L is the (varying) length of a cylindrical shell.
Table 5.2.1 The methods of disks and shells for calculating the volume of a solid of revolution
In each of the following examples, if A is the plane region bounded by the x-axis and the graphs of y=x2 and x=1, use the designated method (disks or shells) to calculate the volume of the solid of revolution formed when A is rotated about the indicated axis.
Disks - about the x-axis.
Disks - about the line y=−1.
Disks - about the y-axis.
Disks - about the line x=2.
Shells - about the x-axis.
Shells - about the line y=−1.
Shells - about the y-axis.
Shells - about the line x=2.
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