星形线x=acos³t,y=asin³t,(0≤t≤2π)所围成图形的面积S=∫π/2→0 asi

Question

如题所述

Answer 1

利用曲线积分计算曲线所围成图形的面积
星形线x=acos³t,y=asin³t,0≤t≤2:
[r(t)]^2=[x(t)]^2+[y(t)]^2=a^2(cost)^6+a^2(sint)^6
=a^2[(cost)^2+(sint)^2][(cost)^4+(sint)^4-(cost)^2(sint)^2]
=a^2[1-3(cost)^2(sint)^2]
所以面积
s=(1/2)∫[r(t)]^2dt
=(1/2)∫(0->2π)
a^2[1-3(cost)^2(sint)^2]dt
=5πa^2/8

Answer 2

由对称性,S＝4∫(0→a)ydx
＝4∫(π/2→0)
a(sint)^3
d[a(cost)^3]
＝12a^2∫(0→π/2)
(sint)^4(cost)^2
dt
＝12a^2∫(0→π/2)
[(sint)^4－(sint)^6]
dt
＝12a^2[3/4*1/2*π/2－5/6*3/4*1/2*π/2]
＝(3πa^2)/8
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