In this section, we will be rotating (spinning) a two-dimensional figure about a line to produce a three-dimensional shape, called a solid of revolution.
A solid of revolution is a three-dimensional figure obtained by rotating a two-dimensional figure (or curve) around a straight line (called the axis) that lies in the same plane.

An example of a rotation about a line is the concept of a revolving door. The panel of one section of the glass door is "pushed" about a center pole, creating an outer cylindrical path around the pole. While the revolving door does not create a "solid", it does illustrate the process of revolution. If the door filled the entire space as it revolved, a cylindrical solid would be formed.
revolving door1

With the idea of revolution (rotation) in mind,
let's start revolving some geometric shapes to create geometric solids.

Let's start by rotating right triangle ACB about a vertical line, as shown below. As the triangle is revolved about the line, the vertices A and C remain stationary, while vertex B follows the path of a circle. As the triangle rotates, ab creates the outline of a cone. Since the triangle's interior is shaded, it creates the interior of the cone, creating a solid figure.

Note: If the 2-D figure (or a portion of the figure) is shaded, then that portion of the 3-D figure will be solid when the revolution occurs.

In this problem, two semicircles are rotated about a vertical line. By the shading, we know that only the portion between the semicircles will create the 3-D solid. The cross-section of the solid shows that it resembles a round gum ball with a hollow center (a solid sphere with an empty sphere in its center).



Note: If the 2-D figure is not tangent (touching) to the line of rotation (axes), then the solid will be in the form of a "ring", with an open section between portions of the solid.

This 2-D figure does not touch the line around which it is to be revolved. As a result, the 3-D solid will have a "ring" appearance (with an open center section).
This particular solid is called a torus, and looks like a dough-nut.

Note: The 2-D figure may appear on coordinate axes, from which you can obtain measurement information about the figure. Rotations will most likely be about the y-axis or the x-axis (but may be about any vertical or horizontal line.)
(Rotations about slanted lines in the coordinate plane will not be discussed at this level.)

A rectangle, shown in the coordinate plane (below, left), is to be rotated about the y-axis. Sketch the resulting solid on the grid and find its volume in cubic units.
The 3-D sketch, which is a cylinder, is shown below on the right.
Using the coordinate grid, the width of the rectangle is 3 units and its height is 4 units.rotateformula4

Note: It may be the case that a portion of a function (or functions) will represent the 2-D figure on coordinate axes to be revolved about an axis or a vertical or horizontal line.
(Rotations about slanted lines in the coordinate plane will not be discussed at this level.)

The function f (x) = x2, on the domain 0 < x < 1, is graphed. The resulting graph is then revolved about the y-axis. Sketch the 3-D solid.
Will the volume of this solid be greater than or less than a cone of radius 1 unit and a height of 1 unit?

The volume of this shape will be slightly greater than the volume of a cone with comparable dimensions. This figure is broader near the vertex than a straight sided cone.


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