利用柱面坐标系求三重积分z=x^2+y^2 z=2y.求∫∫∫Zdv

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  • 该立体投影到xoy面为x²+y²=2y,即Dxy:x²+(y-1)²=1,其极坐标方程为:r=2sinθ

    ∫∫∫zdv

    =∫∫ (∫[0--->2y]zrdz) drdθ

    =∫∫ (∫[0--->2rsinθ]zrdz) drdθ

    =1/2∫∫ z²r |[0--->2rsinθ] drdθ

    =2∫∫ r³sin²θ drdθ

    =2∫[0--->π]∫[0--->2sinθ] r³sin²θ drdθ

    =1/2∫[0--->π] r⁴sin²θ |[0--->2sinθ] dθ

    =8∫[0--->π] sin⁶θ dθ

    =∫[0--->π] (1-cos2θ)³ dθ

    =∫[0--->π] (1-3cos2θ+3cos²2θ-cos³2θ) dθ

    =∫[0--->π] (1-3cos2θ+3/2(1+cos4θ)-sin³2θ) dθ

    =∫[0--->π] (5/2-3cos2θ+3/2cos4θ-sin³2θ) dθ

    =(5/2)θ-(3/2)sin2θ+(3/8)sin4θ-∫[0--->π] sin³2θdθ

    =(5/2)θ-(3/2)sin2θ+(3/8)sin4θ+1/2∫[0--->π] sin²2θd(cos2θ)

    =(5/2)θ-(3/2)sin2θ+(3/8)sin4θ+1/2∫[0--->π] (1-cos²2θ)d(cos2θ)

    =(5/2)θ-(3/2)sin2θ+(3/8)sin4θ+1/2cos2θ-1/6cos³2θ |[0--->π]

    =5π/2