COPYRIGHT NOTICE
Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
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Surface of revolution around x
Surface area of rotational parabolid
1 unit = 200 user units
The origin of the figure is at (0,0)
©
Samuel Dagan
0.5
1.0
1.5
2.0
-1.
-.5
0
0.5
1.0
ΔL
x
y
Surface of revolution around x
The parabola y=√(x/2)
in the interval
0 ≤ x ≤ 2
is going to rotate next by an
angle of 2π
in space about the x axis.
A surface obtained by the
rotation of
a curve, is a surface of revolution.
This
one is called a paraboloid of
revolution.
For each short interval of Δx,
there is a
corresponding length of the curve
ΔL ,
such that ΔL = Δx √[
1+(dy/dx)² ] .
The rotation of such an
element forms
a collar of radius |y(x)| . An
example
of a collar at x = 1
will be drawn next.
In the present example the collar
has
a radius of 1/√2 ,
Δx = 0.1 ,
dy/dx = 1/√8
and ΔL =
0.106...
In the general case, the surface
area of any such collar is
2π|y(x)|ΔL , independently
of the nature of the revolved
function. The summation over all the collars in
the limit
of Δx→0
yields S = 2π∫dx|y|√[1+
(dy/dx)² ]
as the total area of the surface.
The limits of the integral
correspond to the x interval.
In our example S =
9.175...
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