化为极坐标形式的二次积分:∫[0,1]dx∫[0,1]f﹙x,y﹚dy
人气:491 ℃ 时间:2019-08-19 13:11:55
解答
∫[0,1]dx∫[0,1] f(x,y) dy
=∫∫ f(x,y) dxdy 积分区域为矩形:0≤x≤1,0≤y≤1
作y=x将矩形分为两部分分别来做,
x=1对应的极坐标方程为:rcosθ=1,即r=1/cosθ
y=1对应的极坐标方程为:rsinθ=1,即r=1/sinθ
原式=∫∫ f(rcosθ,rsinθ)r drdθ
=∫ [0→π/4] dθ∫[0→1/cosθ] f(rcosθ,rsinθ)r dr+∫ [π/4→π/2] dθ∫[0→1/sinθ] f(rcosθ,rsinθ)r dr
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