利用柱面坐标系求三重积分z=x^2+y^2 z=2y.求∫∫∫Zdv我想了很久了
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![利用柱面坐标系求三重积分z=x^2+y^2 z=2y.求∫∫∫Zdv我想了很久了](/uploads/image/z/11225870-62-0.jpg?t=%E5%88%A9%E7%94%A8%E6%9F%B1%E9%9D%A2%E5%9D%90%E6%A0%87%E7%B3%BB%E6%B1%82%E4%B8%89%E9%87%8D%E7%A7%AF%E5%88%86z%3Dx%5E2%2By%5E2+z%3D2y.%E6%B1%82%E2%88%AB%E2%88%AB%E2%88%ABZdv%E6%88%91%E6%83%B3%E4%BA%86%E5%BE%88%E4%B9%85%E4%BA%86)
利用柱面坐标系求三重积分z=x^2+y^2 z=2y.求∫∫∫Zdv我想了很久了
利用柱面坐标系求三重积分z=x^2+y^2 z=2y.求∫∫∫Zdv
我想了很久了
利用柱面坐标系求三重积分z=x^2+y^2 z=2y.求∫∫∫Zdv我想了很久了
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