直接套平面曲线长度公式:
L=∫ √ [ 1+(y')^2 ] dy (y:a->b)
= ∫ √ [ 1+ ((2/3) y^(-1/3))^2 ] dy (y:1->8)
= ∫ √ [ 1+ (4/9) y^(-2/3) ] dy (y:1->8)
令 y = t^3
= ∫ √ [ 1+ (4/9) t^(-2) ] d(t^3) (t:1->2)
= ∫ t √ ( 9t^2+ 4 ) dt (t:1->2)
= (9t^2+ 4)^(3/2) / 27 (t:1->2)
= [ 40^(3/2)-13^(3/2) ] / 27
= (80√10-13√13) / 27
( ≈ 7.633705416 )