The solid generated by rotating a plane area about an axis in its plane is called a solid of revolution. The volume of a solid of revolution may be found by the following procedures:

## Circular Disk Method

The strip that will revolve is perpendicular to the axis of revolution. In this method, the axis of rotation may or may not be part of the boundary of the plane area that is being revolved.

Using Horizontal Strip

The disk as shown in the figure has an outer radius of xR, a hole of radius xL, and thickness dy. The differential volume is therefore π xR2dy - π xL2dy and the total volume is...

$\displaystyle V = \pi {\int_{y_1}^{y_2}} ({x_R}^2 - {x_L}^2) \, dy$

The integration involved is in variable y since the derivative is dy, xR and xL therefore must be expressed in terms of y. If the axis of revolution is part of the boundary of the plane area that is being revolved, xL = 0, and the equation reduces to...

$\displaystyle V = \pi {\int_{y_1}^{y_2}} {x_R}^2 \, dy$

Using Vertical Strip
From the figure shown below, the volume can be found by the formula...

$\displaystyle V = \pi {\int_{x_1}^{x_2}} ({y_U}^2 - {y_L}^2) \, dx$

If yL = 0, we have

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