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Brain Teaser - Area enclosed by a parabola and a line
09-21-2015, 05:53 PM (This post was last modified: 09-21-2015 09:00 PM by fhub.)
Post: #30
RE: Brain Teaser - Area enclosed by a parabola and a line
(09-18-2015 05:51 PM)Gerson W. Barbosa Wrote:  y=x^4
u: 0.743514882370 Area: 1.229543552881

y=x^6
u: 0.814724747205 Area: 1.119930598753

y=x^8
u: 0.86797579708 Area: 1.24466858824
Hi Gerson,

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:
Code:

n= 2   u=0,500000000   A=1,333333333
n= 4   u=0,743514882   A=1,229543553
n= 6   u=0,814724746   A=1,119930599
n= 8   u=0,850322007   A=1,034513042
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
And here a list from 100 to 1000:
Code:

n= 100   u=0,976938065   A=0,381166646
n= 200   u=0,986757839   A=0,275172584
n= 300   u=0,990499224   A=0,226444741
n= 400   u=0,992515499   A=0,196933121
n= 500   u=0,993789394   A=0,176609553
n= 600   u=0,994672548   A=0,161517493
n= 700   u=0,995323449   A=0,149737975
n= 800   u=0,995824495   A=0,140212382
n= 900   u=0,996222942   A=0,132302458
n=1000   u=0,996547911   A=0,125597430

Franz
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RE: Brain Teaser - Area enclosed by a parabola and a line - fhub - 09-21-2015 05:53 PM



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