Question

高数积分，在线等，挺急得，求详细步骤

Answer 1

分两部分

第一部分，奇函数，对称区间定积分为0；

第二部分就简单了

答案：πR^3/3

 

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不对

追答

哪里不对啊

追问

答案是这个

最后一部怎么算

追答

加绝对值就是偶函数了，稍等

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谢谢

追答

Answer 2

f(θ) = -(1/3)R^3(sinθ)^3
f(-θ) =-f(θ)
∫(-π/2->π/2) [-(1/3)R^3(sinθ)^3 + (1/3)R^3] dθ
=∫(-π/2->π/2) (1/3)R^3 dθ
= (1/3)R^3[θ]|(-π/2->π/2)
=(1/3)πR^3

追问

答案是这个

可是最后一步怎么算的

追答

∫(0->Rcosθ)r.√(R^2-r^2)dr
=(-1/2) ∫(0->Rcosθ)√(R^2-r^2) d(R^2-r^2)
=(-1/3) [(R^2-r^2)^(3/2)](0->Rcosθ)
=(-1/3)( R^3|sinθ|^3 -R^3)

∫(-π/2->π/2)[∫(0->Rcosθ)r.√(R^2-r^2)dr ] dθ
=∫(-π/2->π/2)(-1/3)( R^3|sinθ|^3 -R^3) dθ
=-(1/3)R^3 [ ∫(-π/2->π/2)|sinθ|^3 dθ - π ]
=-(1/3)R^3 [ ∫(0->π/2)(sinθ)^3 dθ -∫(-π/2->0)(sinθ)^3 dθ - π ]
=-(1/3)R^3 [ 2∫(0->π/2)(sinθ)^3 dθ - π ]

consider
∫(0->π/2)(sinθ)^3 dθ
=-∫(0->π/2)[1-(cosθ)^2 ]dcosθ
=-[cosθ - (1/3)(cosθ)^3]|(0->π/2)
=1-1/3
=2/3

∫(-π/2->π/2)[∫(0->Rcosθ)r.√(R^2-r^2)dr ] dθ
=-(1/3)R^3 [2∫(0->π/2)(sinθ)^3 dθ - π]
=-(1/3)R^3 [ 4/3 - π ]
=(1/3)R^3 ( π- 4/3 )

Answer 3

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不对

答案是这个，以后怎么算

最后一步

追答