# Solids of Revolution by Integration

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 x_{R}, a hole of radius x_{L}, and thickness dy. The differential volume is therefore π x_{R}^{2} dy - π x_{L}^{2} dy and the total volume is...

The integration involved is in variable y since the derivative is dy, x_{R} and x_{L} 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, x_{L} = 0, and the equation reduces to...

**Using Vertical Strip**

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

If y_{L} = 0, we have

Where y_{U} and y_{L} are functions of x.

# Cylindrical Shell Method

The strip that will revolve is parallel to the axis of revolution. The volume of revolution is obtained by taking the limit of the sum of cylindrical shell elements, each of which is equal in volume to the mean circumference times the height times the thickness.

**Using Horizontal Strip**

**Using Vertical Strip**