Brain Teaser  Area enclosed by a parabola and a line

09152015, 03:29 AM
(This post was last modified: 09182015 04:19 AM by Gerson W. Barbosa.)
Post: #13




RE: Brain Teaser  Area enclosed by a parabola and a line
(09132015 04:23 PM)Thomas Klemm Wrote: Thus: Things start to get complicated for the biquadratic parabola, if I've done it right: \(y=\frac{4u^{7}+ux}{4u^{3}}\) u: 0.760000000000 Area: 1.23320954839 u: 0.750000000000 Area: 1.23009174283 u: 0.740000000000 Area: 1.22972179352 u: 0.745000000000 Area: 1.22956112013 u: 0.744000000000 Area: 1.22953780108 u: 0.743750000000 Area: 1.22953629609 u: 0.743687500000 Area: 1.22953619031 u: 0.743671875000 Area: 1.22953618081 u: 0.743664062500 Area: 1.22953617857 u: 0.743660156250 Area: 1.22953617806 u: 0.743659179690 Area: 1.22953617801 u: 0.743658691410 Area: 1.22953617799 u: 0.743658203125 Area: 1.22953617797 u: 0.743656250000 Area: 1.22953617799 u: 0.743652798750 Area: 1.22953617828 u: 0.743652343750 Area: 1.22953617836 u: 0.743648437500 Area: 1.22953617913 u: 0.743640625000 Area: 1.22953618202 u: 0.743625000000 Area: 1.22953619273 u: 0.743500000000 Area: 1.22953652222 u: 0.743000000000 Area: 1.22954217016 Dull figures... Hopefully I'm mistaken :) Cheers, Gerson. Edit to include correct values. The table above is wrong, except for u=0.74, due to a silly error in my first Pascal program. I chose a numeric method because it's easy to extend the problem to x^6, x^8, x^10... The result is accurate to 8 digits, but more are possible (Three pairs of values are used to obtain the equation of the parabola that encompasses them, then its minimum is used as a new u). u: 0.730000000000 Area: 1.232203807010 u: 0.740000000000 Area: 1.229721793524 u: 0.741869418802 Area: 1.229582549556 u: 0.743490000000 Area: 1.229543561785 u: 0.743514882370 Area: 1.229543552881 u: 0.743515349953 Area: 1.229543552884 u: 0.743543197140 Area: 1.229543564411 u: 0.744136760955 Area: 1.229549111764 u: 0.75000000000o Area: 1.230145031369 

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