A rectifiable curve is one that has a finite arc length.
A sufficient condition for the graph of a function f to be rectifiable between (a,f(a)) and (b,f(b)) is
that f' be continuous on [a,b]. Such a function is continuously differentiable on [a,b], and
its graph on the interval [a,b] is a smooth curve.
![\[
s = \int ds = \int_a^b \sqrt{1 + f^\prime(x)^2} ~dx
\]](http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/gifs/arclength_12.gif)
If the graph of a continuous function is revolved about a line, the resulting surface is
a surface of revolution.
The area A of the surface of revolution formed by rovolving the graph of f about a horizontal
or vertical axis is
![\[
A = \int_a^b 2\pi f(x) \sqrt{1 + f^\prime(x)^2}~dx
\]](http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/gifs/arclength_33.gif)
where f(x) is the distance between the graph of f and the axis of revolution.
Usually, f(x) is equal to the distance between the graph of f and the axis of revolution,
like this example.
SurfaceOfRevolution(x^3,0..1,
'axis'='horizontal','distancefromaxis'
=0,'output'='plot');
But sometimes r(x) has a different value with f(x) like below problem.
It can be common mistake in solving surface area problems.
SurfaceOfRevolution(x^2,0..2^(1/2),
'axis'='vertical',
'distancefromaxis'=0,'output'='plot');
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