Surfaces of Revolution

1
Surfaces of Revolution

• Cylinders and Cones
• Cone Sections
• Approximating General Surfaces of Revolution
• Area Formula
• The Area of a Sphere

2
Cylinders and Cones
A surface of revolution is obtained by letting
the graph of a function revolve around a line,
usually the x-axis. Cylinders and cones are
simplest surfaces of revolution.
r
r
h
h
3
The Area of a Circular Cylinder
To compute the surface area of a circular
cylinder with radius r and height h, cut the
cylinder along the red line indicated in the
picture below.
The surface area of a circular cylinder of radius
r and height h is 2prh.
Conclude
4
The Area of a Cone
Consider a cone with a circular base of radius r
and height h as indicated in the picture below.
To compute the surface area of the cone, cut it
open along the red line. One gets a section of a
circle.
r
h
The area of the cone is the area of the indicated
section of a disk of radius l.
Conclude
The area of a cone with base radius r and slant
height l is prl.
5
The Area of a Band on a Cone
Consider a cone band as indicated in the picture
below. The black section has been obtained by
letting the blue line segment on the red line
rotate around the x-axis. Let l1 be the slant
height of the cone with circular basis of radius
r1, and let l2 l1 l be the slant height of
the cone with basis of radius r2.
l1
l
r1
r2
Conclude
The area of a cone band with average radius r
and slant width l is 2prl.
6
The Area of a Section of a Cone
Conclude
The area of a cone band with average radius r
and slant width l is 2prl.
l1
l
r1
r2
7
The Area of a Surface of Revolution
Letting the graph of a non-negative function f
rotate around the x-axis we get a surface of
revolution.
Idea
We approximate this surface of revolution by
certain cone sections and compute the total area
of these cone sections. That gives a Riemann sum
for an integral. This integral is the area of
the surface of revolution.
8
The Area of a Surface of Revolution
Union of the blue cone sections approximates the
surface of revolution.
9
The Area of a Sphere
Formula
Example
The Area of a Sphere of Radius r.
Conclude
The area of a sphere of radius r is 4pr2.