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1. Finding volume of a solid of revolution using a disc method.
2. Finding volume of a solid of revolution using a washer method.
3. Finding volume of a solid of revolution using a shell method.
If a region in the plane is revolved about a given line, the resulting solid is a solid of revolution, and the
line is called the axis of revolution. When calculating the volume of a solid generated by revolving a
region bounded by a given function about an axis, follow the steps below:
1. Sketch the area and determine the axis of revolution, (this determines the variable of integration)
2. Sketch the cross-section, (disk, shell, washer) and determine the appropriate formula.
3. Determine the boundaries of the solid,
4. Set up the definite integral, and integrate.
1. Finding volume of a solid of revolution using a disc method.
The simplest solid of revolution is a right circular cylinder which is formed by revolving a rectangle about
an axis adjacent to one side of the rectangle, (the disc).
To see how to calculate the volume of a general solid of revolution with a disc cross-section, using
integration techniques, consider the following solid of revolution formed by revolving the plane region
bounded by f(x), y-axis and the vertical line x=2 about the x-axis. (see Figure1 to 4 below):
Figure 1. The area under f(x), bounded by f(x), x-axis, Figure 2. Basic sketch of the solid of revolution
y-axis and the vertical line x=2 rotated about x-axis with few typical discs indicated.
Figure 3. Family of discs Figure 4. The 3-D model of the solid of revolution.