用极坐标法计算二重积分∫∫x^2/y^2dxdy D:x=2,y=x,xy=1所围成区域
问题描述：
用极坐标法计算二重积分∫∫x^2/y^2dxdy D:x=2,y=x,xy=1所围成区域
答
积分区域：arctan(1/4)《θ《π/4 √2/sin2θ《r《2/cosθ
∫∫x^2/y^2dxdy
=∫(arctan(1/4),π/4)dθ∫(√2/sin2θ,2/cosθ)(cosθ/sinθ)^2rdr
=(1/2)∫(arctan(1/4),π/4)(cosθ/sinθ)^2(2/(sin2θ)^2-4/(cosθ)^2)dθ
= (1/2)∫(arctan(1/4),π/4)(1/(sinθ)^2(1/2(sinθ)^2-4)dθ
= (1/2)[(1/2)(-1/3)cotx((cscx)^2+2)+4cotx)|(arctan(1/4),π/4)
以下代值,自己试试