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∫(R^2-y^2)^(3/2) dy
=y(R^2-y^2)^(3/2)-∫y*(3/2)*[(R^2-y^2)^(1/2)]*(-2y)dy
=y(R^2-y^2)^(3/2)+3∫y^2*[(R^2-y^2)^(1/2)]dy
...(1)
令y=RsinA,A属于[-pi/2,pi/2]
则∫y^2*[(R^2-y^2)^(1/2)]dy
=∫R^2[(sinA)^2]*RcosA*d(RsinA)
=∫R^4[(sinAcosA)^2]dA
=∫R^4[(sin2A)^2/4]dA
=∫(R^4)*[(1-cos4A)/8]dA
=(A-0.25*sin4A)*R^4/8+C
=[arcsin(y/R)]-{y*[(R^2-y^2)^(3/2)]/R^4}+C...(2)
将(2)代入(1)得:∫(R^2-y^2)^(3/2) dy
=y(R^2-y^2)^(3/2)+[arcsin(y/R)]-{y*[(R^2-y^2)^(3/2)]/R^4}+C
1年前
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