To find surfaces of same area slices

[gudi]

In an attempt to find all surface of revolution with meridian y(x)
and slice width (b-a) perpendicular to x-axis

of symmetry, it is now required to solve:

Solve [ Integrate[ y[x] * Sqrt[1 + y'[x]^2 ] , { x, a, b} ] ==( b -
a) , y, x ]

a, b are constants, a < b < 1. Primes with respect to x.

y [x] = 1, Sqrt [1 - x^2] are two known solutions.( unit radius
cylinder and sphere ) -- known from Archimedes

times, some twenty two centuries ago.

Best Regards,
Narasimham

[gudi]

Obvious solutions obtained by setting integrand = 1 are (x-c)^2 + y^2
= 1 which are unit circles on x-axis and,
may be also their envelope y = +/- 1, but the question is, is it the
only solution?

Narasimham

[Ray Vickson]

On Mar 20, 3:59 pm, gudi <mathm...@hotmail.com> wrote:
> In an attempt to find all surface of revolution with  meridian y(x)
> and slice width (b-a) perpendicular to x-axis
>
> of symmetry, it is now required to solve:
>
> Solve [ Integrate[ y[x] * Sqrt[1 + y'[x]^2 ]   , { x, a, b} ] ==( b -
> a) , y, x ]

Please do not use Mathematica notation; not everybody uses that
software. Certainly, not everybody has access to it, or wants to. Just
use standard notation.

>
> a, b are constants, a < b < 1. Primes with respect to x.
>
> y [x] = 1, Sqrt [1 - x^2] are two known solutions.( unit radius
> cylinder and sphere ) --  known from  Archimedes
>
> times, some twenty two centuries ago.
>
> Best Regards,
> Narasimham

If I understand your question correctly, you want to know the most
general solution f(x) of the equation
int_{x=a..b} f(x)*sqrt(1 + f'(x)^2) dx = b-a. If you take any
function g(x) such that int+{x=1..b} g(x) dx = b-a, then to get f(x)
you need to solve f*sqrt(1+f'^2) = g, or f'(x) = +-sqrt(g(x)^2 -
f(x)^2)/f(x). If you can solve this differential equation, you have
f(x).

R.G. Vickson

[gudi]

Yes, you got it ok, you say set the integrand equal to g(x), so that

int_{x = a..b} g(x) dx  = b-a

Just to test a simple case I took g(x) = x^2 + 1, a = 0, y(a) = 1,
y'(0) = 0 for a numerical solution (as the analytical solution
requires inverting/changing of axes etc. to solve it, cumbersome).

The numerical solution plot looks somewhat like an ellipse of semi-
diameters 1 and ~0.69547, in the first quadrant.

y co-ords evaluated on x-axis at 0.1 intervals are :
(1,0.98994,0.959046,0.904900,0.8224169,0.7008428,0.511678).

but their first differeces do _not_ remain constant, which does occur
for a sphere and cylinder.

The motivation in my post was to seek to find whether in int[{ x=a,b}
y* sqrt(1 + y'^2) dx = (b-a), any theorem of definite integrals exists
that enables situations other than the obvious equating of integrand
g(x) = y* sqrt(1 + y'^2) to 1.( which has solutions y = 1, (x -c)^2 +
y^2 = 1).

Regards
Narasimham