Method

Axis

Integral

Comments

Disks

Horizontal

Vertical

$\mathrm{\π} {\int }_a^b{\mathrm{\ρ}}^2\left(x\right) \mathit{\ⅆ}x$ 

$\mathrm{\π} {\int }_c^d{\mathrm{\ρ}}^2\left(y\right) \mathit{\ⅆ}y$ 

$\mathrm{\ρ}$ is the (varying) radius of rotation, the distance from the axis of rotation $\mathrm{\λ}$, to the boundary of the rotated region $A$.

Washers
(Disks with holes)

Horizontal

Vertical

$\mathrm{\π} {\int }_a^b\left(R^2\left(x\right)-r^2\left(x\right)\right) \mathit{\ⅆ}x$ 

$\mathrm{\π} {\int }_c^d\left(R^2\left(y\right)-r^2\left(y\right)\right) \mathit{\ⅆ}y$

$R$ is the radius of rotation of the outer boundary of $A$, whereas $r$ is the radius of rotation of the inner boundary.

Shells

Vertical

Horizontal

$2 \mathrm{\π} {\int }_a^b\mathrm{\ρ}\left(x\right) L\left(x\right) \mathit{\ⅆ}x$

$2 \mathrm{\π} {\int }_c^d\mathrm{\ρ}\left(y\right) L\left(y\right) \mathit{\ⅆ}y$

$\mathrm{\ρ}$ 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

Example 5.2.1

Disks - about the $x$-axis.

Example 5.2.2

Disks - about the line $y\=-1$.

Example 5.2.3

Disks - about the $y$-axis.

Example 5.2.4

Disks - about the line $x\=2$.

Example 5.2.5

Shells - about the $x$-axis.

Example 5.2.6

Shells - about the line $y\=-1$.

Example 5.2.7

Shells - about the $y$-axis.

Example 5.2.8

Shells - about the line $x\=2$.