Re: [微分]質心的小問題
※ 引述《chingyes (axsazxc)》之銘言：
: suppose that (X爸.Y爸)is the centroid of region bounded by the graph of the
: parametric equations x=t-sint ,y=1-cost ,0<t<拍 and the x-axis
: Find Y爸=?
就代公式阿
pi
Y爸=2^(1/2)*S (1-cost)^(3/2) dt
0
pi
=2^(1/2)*2^(3/2)*S [(1-cost)/2]^(3/2) dt ... 半角公式
0
pi
=4*S sin^3(t/2) dt
0
令u=t/2,dt=2du
pi/2
=8*S sin^3u du
0
再令w=cosu,dw=-sinudu,sin^2u=1-w^2
1
=8*S (1-w^2)dw
0
1
=8*[w-(w^3/3)]｜
0
=8[(2/3)-0]
=16/3
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