求星形线x = a cos3 t ,y = a sin 3 t 的全长
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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