Computer Algebra Recipes

186 CHAPTER 4. NONLINEAR ODE MODELS
date the shapes of its most beautiful (by Erehwon standards) female citizens.
Being an amateur scientist on the side and also to keep down material cost, he
wantsthesurfaceareaofclothintheone-piecedresstobeaminimum. The
smaller upper ring at x = 0 is of radius y = 1 sretem (the local unit of length),
and the lower, larger, ring at x = 1 sretem is of radius y = 2 sretem. To answer
the question of what the shape of the surface should be to minimize the area,
Gino consults one of Erehwon's most prominent mathematicians, Greg Arious
Nerd, who says that this problem was solved many millennia ago. Greg then
proceeds to outline the solution to the problem for his dress-designing friend.
For the bene¯t of readers who may not understand the peculiar Northern ErehwonesedialectspokenbyGreg,weshalltranslatewhathehastosay.
The
authors are not totally °uent in this language, so bear with us if we occasionally
are in doubt about the exact words and thus resort to paraphrasing what Greg
says instead of reporting his remarks verbatim.
\It's all very simple," Greg starts out. \The surface area of the dress can be
generated by revolving the curve y(x) aboutthex-axis. So what 5 is the shape
of y(x)?
ds
y
y(x)
Figure 4.12: Determining the surface area of the dress.
Referring to Figure 4.12, an element dA of surface area is given by
dA =(2¼y) ds =2¼y p (dx) 2 +(dy) 2 =2¼y p 1+(y 0 ) 2 dx; (4.15)
where the prime indicates a derivative with respect to x. So the surface area
between x =0andx =1is
A =2¼
Z 1
0
y p 1+(y 0 ) 2 dx: (4.16)
5Author's note: This appears to be a rhetorical question on Greg's part, since Gino hasn't
opened his mouth yet.
x