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 rotating a parabola about its axis of symmetry.

• Paraboloid: The surface generated by rotating the parabola y = x^2 (or z = x^2 + y^2 in three dimensions) about its 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 a paraboloid, highlighting its bowl-like shape and symmetry.

• The second image (image_2) shows the same solid with coordinate axes, emphasizing the axis of revolution (typically the z-axis or y-axis).

Volume of a Paraboloid by Integration

The volume of a solid of revolution can be found using the disk method or washers method. For a paraboloid formed by rotating y = x^2 about the y-axis from y = 0 to y = h:

• The radius at height y is r = sqrt(y).

• The area of a cross-sectional disk is A(y) = πr^2 = πy.

• The volume is given by the integral:

• This formula gives the volume of a paraboloid of height h generated by rotating y = x^2 about the y-axis.

Key Concepts and Definitions

• Solid of Revolution: A 3D object formed by rotating a 2D curve about an axis.

• Disk Method: Used when the cross-sections perpendicular to the axis of revolution are disks.

• Washer Method: Used when the cross-sections are washers (disks with holes).

• Paraboloid: The surface or solid generated by rotating a parabola about its axis.

Example: Volume of a Paraboloid

Example: Find the volume of the solid formed by rotating y = x^2 about the y-axis from y = 0 to y = 3.

• Radius at height y: r = sqrt(y)

• Area: A(y) = πy

• Volume:

Summary Table: Methods for Volumes of Revolution