前>>355
CとLの距離をpとおくと、
回転体の体積=2πだから、
x≧0において、
(1/2)V_CL=π
=π∫[t=0→√(1-p^2)]｛p+√(1-t^2)｝^2dt+2π∫[t=√(1-p^2)→1]
t= sinαとおくとdt=cosαdα
(1/2)V_CL=π
=π∫[α=0→θ](p+cosα)^2cosαdα+2π∫[α=θ→π/2]cosαcosαdα
=π∫[α=0→θ](p^2cosα+2pcos^2α+cos^3α)dα+2π∫[α=θ→π/2](1/2+cos2α)dα
=π∫[α=0→θ]｛p^2cosα+p(1+cos2α)+cos3α/4+3cosα/4｝dα+2π∫[α=θ→π/2](1/2+cos2α)dα
=π∫[α=0→θ]｛(p^2+3/4)cosα+p+pcos2α+cos3α/4｝dα+π∫[α=θ→π/2](1+cos2α)dα
=π[α=0→θ][(p^2+3/4)sinα+pα+psin2α/2+sin3α/12]+π[α=θ→π/2][α+sin2α/2]
=π｛(p^2+3/4)sinθ+pθ+psin2θ/2+sin3θ/12｝+π(π/2-θ- sin2θ/2)
=π｛(p-1)θ+π/2+(p^2+3/4)sinθ+(p-1)sin2θ/2+sin3θ/12｝
=π｛(p-1)θ+π/2+(p^2+3/4)sinθ+(p-1)sinθcosθ+sinθ/4-sin^3θ/3｝
=π｛(p-1)θ+π/2+(p^2+1)sinθ+(p-1)sinθcosθ-sin^3θ/3｝
=π｛(cosθ-1)θ+π/2+(cos^2θ+1)sinθ+(cosθ-1)sinθcosθ-sin^3θ/3｝
=π｛(cosθ-1)θ+π/2+2cos^2θsinθ+sinθ-sinθcosθ-sinθ(1-
cos^2θ)/3｝
=π｛(cosθ-1)θ+π/2+7cos^2θsinθ/3+2sinθ/3-sinθcosθ｝
(cosθ-1)θ+π/2+7cos^2θsinθ/3+2sinθ/3-sinθcosθ=1
(1/3)(7cos^2θ-3cosθ+2)sinθ=1-π/2+(1-cosθ)θ
2(7cos^2θ-3cosθ+2)sinθ=3｛2(1-cosθ)θ-(π-2)｝
θ=
∴p=cosθ=
θのだいたいの大きさはわかります。
pは0.2〜0.25ぐらいじゃないかと。