This exercise considers computing the curve that produces a surface of revolution with minimum area. Given two points ( ) and ( ) in the plane, with and  and  positive, assume that ( ) is a smooth,...

This exercise considers computing the curve that produces a surface of revolution with minimum area. Given two points () and () in the plane, with

and
 and
 positive, assume that
() is a smooth, positive, function connecting these points. If
() is rotated about the
axis, the resulting surface has area

It should be noted that the curve is required to satisfy() =
 and
() =
. Also, unlike the brachistochrone problem,
() does not have a singularity(at either end).(at either end).

(a) Suppose the grid points are

₀
= 0,

₁
=
,

₂
= 2, ··· ,


_n+1
= 1, where
 = 1/( + 1). Writing
write down the composite trapezoidal approximation for
.

(b) Use a centered second-order approximation for
’ at

₁,

₂, ··· ,


_n
, and a first-order approximation for
’ at

₀
and


_n+1. Using these with the result from part (a), what is the resulting approximation for
?

(c) The minimum of
 from part (b) can be found using the MATLAB command
where
 is the approximation from part (b) and
Y
is an
-vector containing a starting guess for (
₁,

₂, ··· ,


_n
)
^T

. What would be a good, simple, and nonzero choice for
Y, and why is it a good choice?

(d) Taking
 = 0,
 = 1,
 = 1, and
 = 3, plot the numerical solution for
() for
 = 4,
 = 9, and
 = 19.

(e) In calculus it is shown that the curve producing the minimum area is
 = (1/)cosh(+
), where
 and
 are determined from the requirements that
() =
 and
() =
. Determine the nonlinear equations that must be solved to find
 and
. Discuss the numerical difficulties associated with solving these equations, compared to the direct solution in parts (a)–(c).

Note: The minimal surface problem has some interesting mathematical complications, and for more about this see Oprea [2007].