∫（-R,R）dx ∫（-根号R^2-x^2,根号R^2-x^2）dy ∫-根号R^2-x^2-y
∫（-R,R）dx ∫（-根号R^2-x^2,根号R^2-x^2）dy ∫-根号R^2-x^2-y
∫（-R,R）dx ∫（-根号R^2-x^2,根号R^2-x^2）dy ∫-根号R^2-x^2-y^2,0）（x^2+y^2）柱坐标球坐标表示并计算.
赞 dongmiaojun 幼苗
共回答了17个问题采纳率：94.1% 举报
原式Ⅰ=∫dθ∫r^3dr∫dz (作柱面坐标变换)
=2π∫r^3√(R^2-r^2)dr
=π∫[(R^2-r^2)^(3/2)-R^2(R^2-r^2)^(1/2)]d(R^2-r^2)
=π[(2/3)R^5-(2/5)R^5]
=(4/15)πR^5.
=2π∫r^3√(R^2-r^2)dr
=π∫[(R^2-r^2)^(3/2)-R^2(R^2-r^2)^(1/2)]d(R^2-r^2)
=π[(2/3)R^5-(2/5)R^5]
=(4/15)πR^5.
1年前 追问
7
r^2(R^2-r^2)^(1/2)=[R^2-(R^2-r^2)](R^2-r^2)^(1/2) =-[(R^2-r^2)^(3/2)-R^2(R^2-r^2)^(1/2)]
可能相似的问题