Calculus

Integration/Applications of Integration2011. 6. 1. 15:33

Arc Length and Surfaces of Revolution

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.

$s = \int ds = \int_a^b \sqrt{1 + f^\prime(x)^2} ~dx$

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

$A = \int_a^b 2\pi f(x) \sqrt{1 + f^\prime(x)^2}~dx$

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.