Volumes of Solids of Revolution

Solids of Revolution and the Paraboloid

In calculus, a solid of revolution is a three-dimensional object obtained by rotating a two-dimensional curve around an axis. One of the most common examples is the paraboloid, which is generated by revolving a parabola about its axis of symmetry.

• Paraboloid: The surface generated by revolving the parabola y = x^2 (or similar) about the y-axis (or x-axis).

• Applications: Paraboloids appear in physics, engineering, and geometry, such as in satellite dishes and reflective surfaces.

The images above illustrate a paraboloid as a solid of revolution:

• The first image (image_1) shows a perspective view of the paraboloid, highlighting its bowl-like shape.

• The second image (image_2) shows the paraboloid with coordinate axes, emphasizing the axis of revolution.

Volume of a Solid of Revolution: Disk and Washer Methods

To compute the volume of a solid of revolution, calculus uses the disk method or the washer method:

• Disk Method: Used when the solid has no hole (e.g., revolving y = f(x) about the x-axis).

• Washer Method: Used when the solid has a hole (e.g., revolving the region between two curves).

Formula (Disk Method):

• If the region under y = f(x) from x = a to x = b is revolved about the x-axis, the volume is:

Example: Paraboloid of Revolution

• Suppose we revolve y = x^2 from x = 0 to x = r about the y-axis. To express x in terms of y, note that x = \sqrt{y}.

• The volume is:

Key Points:

• The cross-sections perpendicular to the axis of revolution are circles (disks).

• The radius of each disk is determined by the function being revolved.

• Integrating the area of these disks along the axis gives the total volume.

Summary Table: Methods for Volumes of Revolution

Example Application: Calculating the volume of a satellite dish, bowl, or reflector shaped as a paraboloid.