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From: Vladimir Bondarenko on 10 Jul 2010 13:50
An exact 1-D integration challenge - 65 - (sqrt, cos)

Hello,

int(cos(((sqrt(4+9*z^2)-3*z)/2)^(1/3)-
((sqrt(4+9*z^2)+3*z)/2)^(1/3)), z= 0..1);

?
Cheers,

Vladimir Bondarenko

Co-founder, CEO, Mathematical Director

http://www.cybertester.com/ Cyber Tester Ltd.

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"...... the challenges imply that a solution is built within the
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From: us on 10 Jul 2010 14:22
Vladimir Bondarenko <vb(a)cybertester.com> wrote in message <9ad26ebd-e29b-4b6d-9c99-a0189b764707(a)d37g2000yqm.googlegroups.com>...
> An exact 1-D integration challenge - 65 - (sqrt, cos)
>
> Hello,
>
> int(cos(((sqrt(4+9*z^2)-3*z)/2)^(1/3)-
> ((sqrt(4+9*z^2)+3*z)/2)^(1/3)), z= 0..1);

a hint:
- this produces an error...
- you must use correct ML syntax...

us
From: David Rusin on 10 Jul 2010 17:46
In article <9ad26ebd-e29b-4b6d-9c99-a0189b764707(a)d37g2000yqm.googlegroups.com>,
Vladimir Bondarenko <vb(a)cybertester.com> wrote:

>int(cos(((sqrt(4+9*z^2)-3*z)/2)^(1/3)-
> ((sqrt(4+9*z^2)+3*z)/2)^(1/3)), z= 0..1);

You're trying to integrate cos(x) dz where x is an algebraic
function of z. You can do this by looking for an antiderivative
of the form a cos(x) + b sin(x) where a, b are functions of
x and z .

In this case, the algebraic relationship between x and z is
x^3 + 3 x + 3 z = 0
which is obviously very easily solved for z, so why not do a
change of variables? The integral is simply int_x0 ^x1 (1+x^2) cos(x) dx
where x0 and x1 are the values of x that correspond to the endpoints
z0 = 0 and z1 = 1 . This integral has antiderivative
2x cos(x) + (x^2-1) sin(x)
which we simply evaluate at the endpoints x = x1 and x = x0 (=0).
Obviously x1 can be given in closed form via the cubic formula.

The complexity of the answer -- and, I guess, the difficulty of finding it --
depends heavily on the genus of the algebraic curve defined by the
relationship between x and z . This particular problem is easy
because the curve has genus 0, i.e. is parameterizable, so we can
rewrite the integral in terms of rational functions of the parameter
(in this case, x).

dave

From: JCH on 11 Jul 2010 13:27


"Vladimir Bondarenko" <vb(a)cybertester.com> schrieb im Newsbeitrag
news:9ad26ebd-e29b-4b6d-9c99-a0189b764707(a)d37g2000yqm.googlegroups.com...
> An exact 1-D integration challenge - 65 - (sqrt, cos)
>
> Hello,
>
> int(cos(((sqrt(4+9*z^2)-3*z)/2)^(1/3)-
> ((sqrt(4+9*z^2)+3*z)/2)^(1/3)), z= 0..1);
>
> ?


Gauss Quadrature:

Integral = 0.876731103691882


JCH



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