求星形线x=acos3t,y=asin3t的全长

Answer 1

求星形线x=a cos³t ,y=asin³t 的全长
星形线关于两个坐标轴和原点都对称,因此只需求出[0,π/2]内之长,再乘以4即
得全长.
dx/dt=-3acos²tsint；dy/dt=3asin²tcost；
L=【0,π/2】4∫√[(-3acos²tsint)²+(3asin²tcost)²]dt
=【0,π/2】12a∫√(cos⁴tsin²t+sin⁴tcos²t)dt
=【0,π/2】12a∫√(cos²tsin²t)dt
=【0,π/2】12a∫costsintdt
=【0,π/2】12a∫sintd(sint)
=12a[(sin²t)/2]【0,π/2】
=6a