Disc Method

Disk integration is a means of calculating the volume of a solid of revolution, when integrating along the axis of revolution. This method models the generated 3 dimensional shape as a stack of an infinite number of disks (of varying radius) of infinitesimal thickness.

To find the volume of a solid of revolution with the disc method, use one of the following.

• Horizontal axis of revolution

• Vertical Axis of Revolution

"Hollow" solid of revolutiona solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis) that lies on the same plane. Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. To apply these methods, it is easiest to draw the graph(s) in question, identify the area that is actually being revolved about the axis of revolution, and then draw a straight line, vertical (parallel to the y-axis) for functions defined in terms of x and horizontal (parallel to the x-axis) for functions defined in terms of x, which is referred to as a slice. The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating along the axis of revolution.

The method can be visualized by considering a thin vertical rectangle at x between y = f(x) on top and y = g(x) on the bottom, and revolving it about the x-axis; it forms a ring (or disc in the case that g(x) = 0), with outer radius f(x) and inner radius g(x). Summing up all of the areas along the interval gives the total volume. Alternatively, where each disc has a radius of f(x), the discs approach perfect cylinders as their height dx approaches zero. Suppose we are given an area bounded by certain curves in the xy-plane. That area is rotated around an axis to form a solid. Each point of the figure is rotated in a circle. If you slice the solid perpendicular to the axis you will see a cross section (the area revealed by the slice) that is either a full disk or a washer (the area between two circles. It's easy to calculate the area of a circle and you can imagine the solid as made of a lot of very thin circles.