20/11/25 15:27:29.84 uNm3BuF0.net
0<a≦1 とする。
放物線を
y = (3/4aa)(a^2 - x^2),
とすれば
y ' = - (3/2aa)x,
L(a) = ∫[0,a] √{1 + (y ')^2} dx
= (3/4)√{1+(2a/3)^2} + (aa/3)arcsinh(3/2a)
= (3/4)√{1+(2a/3)^2} + (aa/3)log((3/2a) + √{1+(3/2a)^2})
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a 放物線 円弧(+線分)
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0.0 0.75 0.50 + 50.0%
0.1 0.76301 0.57854 + 31.9%
0.2 0.79280 0.65708 + 20.7%
0.3 0.83423 0.73562 + 13.4%
0.4 0.88459 0.81416 + 8.65%
0.5 0.94211 0.89270 + 5.53%
0.6 1.00544 0.97124 + 3.52%
2/π 1.02989 1.00 + 2.99%
0.7 1.07359 1.04982 + 2.26%
0.8 1.14574 1.12899 + 1.48%
0.9 1.22127 1.20928 + 0.99%
1.0 1.29964 1.29095 + 0.67%
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