Question

二重积分用极坐标计算怎么画图

Answer 1

对应y=√( 2x-x^2)是一个以(1,0)为圆心,半径为1的半圆,化成极坐标为r=2cosθ,

而y=x是与X轴夹角45度经过原点的直线,

积分区间,0<=r<=2cosθ, π/4<=θ<=π/2,

I=∫ [π/4,π/2] dθ ∫ [ 0,2cosθ] √[(rcosθ)^2+rsinθ)^2]*rdr

=∫ [π/4,π/2] dθ ∫ [ 0,2cosθ] r^2dr

=∫ [π/4,π/2] (r^3/3) dθ  [ 0,2cosθ]

=(1/3)∫ [π/4,π/2] 8(cosθ)^3dθ

=(8/3)∫ [π/4,π/2] [1-(sinθ)^2]d(sinθ)

=(8/3)[sinθ-(sinθ)^3/3][π/4,π/2]

=(8/3)[1-√2/2-(1/3-2√2/24)]

=16/9-10√2/9.