Rotating Areas to Generate Solids of Revolution: Volume.

Having an adult by your side. Have a drill. Take the bit and attach a rectangle by one of the sides parallel to the bit. Use tape to hold it to the bit, so that you have a "flag." Reintroduce the bit into the drill and press the trigger. What do you see? You will notice that when the rectangular area rotates a that high number of revolutions per minute, they leave a visual trail that our eye does not refresh as soon as the area is rotating and no longer in the space were it was. That gives the visual impression of a cylinder. Now detach it and attach it back with two rigid strings at the top and bottom of the sides perpendicular to the bit. Repeat again pressing the trigger, what do you see? Again you will see a cylinder, but this time it is a hollow cylinder or a cylinder with a volume removed along the sides of the bit. You may now repeat with a triangle, a semicircle, etc... to generate other solids of revolution. Using what you learn about finding the volumes of solids. You may brake these rotating areas into rectangles along the distance of the bit. Then using the height of the rectangles as the radius you apply the formula of the circumpherence and you obtain the volume of a cylinder or with this height using it again as the radius you apply the formula of the volume of the cylinder with the base of the rectangle as the height of the cylinder to get the same volume as before. Then you add all those cylinder volumes to get the total volume of the solid generated by rotating the area.

In the industry, the way complicated pipe connectors are obtained is using this process. They subtract the area under the curve of one curve from the area under the curve of other curve. Then brake it into equal width rectangles that in turn are rotated and the resulting volumes of the generated "donnas" are added to get the full volume of the complex pipe connector. Of course, they do it using integrals as you may learn in calculus. But the process is basically the same. This is what we are going to practice in this lesson. Have a calculated adventure!

Lesson's Content

Study this lesson and the corresponding for Surface Area for Solids of Revolution. Then attempt this problem. Scroll down the page for a partial solution after you attempted.

Real World Follow up application


Lesson's Glossary

It is a 3-D figure with circular base, a vertex not in the plane of the circle, and has a curved surface connecting the base with the vertex. The altitude of the cone is the perpendicular segment that goes from the vertex to the plane containing the circle at the base. The height is the length of the altitude. The slant height is the length of the distance from the vertex to the edge of the base. For a right cone it is necessary that the altitude contains the center of the circle at the base.

Conic solid
The set of points between a point (the vertex) and a non-coplanar region (the base), including the point and the region.

The union of the bases and the lateral surface.

Cylindrical surface
The surface of a cylindrical solid whose base is a circle.

Half of a sphere.

Lateral area.

Lateral area
The area of the lateral surface of a solid.

Platonic solid
A Platonic solid is defined as a solid in which all of its faces are congruent regular polygons and the same number of regular polygons meet at each vertex.

It is a polyhedron that has two congruent and parallel faces, called bases. The remaining faces, which are parallelograms are called lateral faces. The altitude is the perpendicular segment whose endpoints are at the bases. The length of the altitude is the height of the prism. A right prism is one with all lateral faces rectangles. An oblique rectangle has some to be nonrectangular. 

It is a polyhedron whose base is a polygon and whose lateral faces are triangles with a common vertex called the vertex of the pyramid. The perpendicular segment whose endpoints are at the base and at the vertex of the pyramid is the altitude, which is the height of the pyramid. A regular pyramid has congruent isosceles triangles in the lateral faces and a regular polygon in the base. The length of the height for the lateral triangles is the slant height for the pyramid.

Regular pyramid
A pyramid whose base is a regular polygon and whose vertex forms a segment with the center of the polygon perpendicular to its plane.

Slant height
The length of a lateral edge of a conic solid.

Right prism
A prism whose direction of sliding is perpendicular to the plane of the base.

A sphere is properly defined as the set of points in the space that are equidistant of one point called center. Planes that go through the center of the sphere, intersect GREAT CIRCLES. The circumference of the sphere is the perimeter of any of its great circles.

The boundary of a 3-D figure.

Surface area
The total area of the surface of a solid.

Unit cube
Unit of measuring volume.



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Partial solution to the featured problem. Showing first a non-accurate logitudinal cut of the bottle, and a second solution with a more accurate longitudinal cut. You may find useful to pause the video from time to time to study particular frames.