作业帮求解答,高数题,如图,谢谢
来源：学生作业帮 编辑：作业帮 分类：综合作业 时间：2024/10/05 04:12:26
求解答,高数题,如图,谢谢
这题楼上太繁琐.用极坐标,注意0《r《Rcosθ,代入得：
=∫(-π/2,π/2)dθ∫(0,Rcosθ)r√(R^2-r^2)dr
=(1/2)∫(-π/2,π/2)dθ∫(0,Rcosθ)√(R^2-r^2)dr^2
=(-1/3)∫(-π/2,π/2)(R^2-r^2)^(3/2)|(0,Rcosθ)dθ
=(1/3)∫(-π/2,π/2)R^3(1-[(sinθ)^2^(3/2)dθ (偶函数在对称区间的积分为半区间的2倍）
=(2R^3/3)∫(0,π/2)(1-(sinθ)^3)dθ (∫(0,π/2)(sinθ)^3)dθ用公式=2/3)
=(2/3)R^3(π/2-2/3)
=∫(-π/2,π/2)dθ∫(0,Rcosθ)r√(R^2-r^2)dr
=(1/2)∫(-π/2,π/2)dθ∫(0,Rcosθ)√(R^2-r^2)dr^2
=(-1/3)∫(-π/2,π/2)(R^2-r^2)^(3/2)|(0,Rcosθ)dθ
=(1/3)∫(-π/2,π/2)R^3(1-[(sinθ)^2^(3/2)dθ (偶函数在对称区间的积分为半区间的2倍）
=(2R^3/3)∫(0,π/2)(1-(sinθ)^3)dθ (∫(0,π/2)(sinθ)^3)dθ用公式=2/3)
=(2/3)R^3(π/2-2/3)