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Brain Teaser - Area enclosed by a parabola and a line
09-21-2015, 06:40 PM (This post was last modified: 09-21-2015 06:53 PM by Gerson W. Barbosa.)
Post: #31
RE: Brain Teaser - Area enclosed by a parabola and a line
(09-21-2015 05:53 PM)fhub Wrote:  since your Area result for x^8 looked rather illogical for me (IMO the area should continuously decrease for higher exponents n), I've written a small program (in TurboPascal) for this problem.
The left intersection point is of course calculated numerically (no exact value possible for n>4), but the integral is calculated exactly (i.e. no numerical method), and the results really confirm my assumption of continuously decreasing areas:
in fact for n->inf the results are: u->1 and A->0

Here's the list for n=2..20:

n= 2   u=0,500000000   A=1,333333330
n= 4   u=0,743514882   A=1,229543550
n= 6   u=0,814724746   A=1,119930600
n= 8   u=0,850322007   A=1,034513040
n=10   u=0,872567780   A=0,966429043
n=12   u=0,888129582   A=0,910591797
n=14   u=0,899777205   A=0,863713409
n=16   u=0,908898125   A=0,823611190
n=18   u=0,916276056   A=0,788780129
n=20   u=0,922392221   A=0,758146096

Hello Franz,

Thanks again for pointing out yet another wrong result of mine. Thanks also for providing solutions to this plentiful of cases.
Since my results for y=x^6 are correct within the accuracy I stated, I surely have made a mistake in the function submitted to the numerical integration and solver for case y=x^8 (I have yet to find it out).
Yes, there's no need to use a numerical integrator as the functions are quite easy to integrate [ integral (x^n)dx = x^(n+1)/(n+1) + k, n ≠ -1 ], only the left integration limits being the complicated parts.
I've found that Derive 4.11 is installed in my HP-200LX and tried some numerical integrations (very slow on the 200LX, BTW). But I haven't explored it much as it is not so intuitive and didn't find a manual.

Best regards,

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RE: Brain Teaser - Area enclosed by a parabola and a line - Gerson W. Barbosa - 09-21-2015 06:40 PM

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