二重积分用极坐标计算怎么画图
来源:学生作业帮 编辑:作业帮 分类:综合作业 时间:2024/08/23 10:09:11
二重积分用极坐标计算怎么画图
![](http://img.wesiedu.com/upload/f/fc/ffcec6da4492c66099ef076537c2fcfa.jpg)
![](http://img.wesiedu.com/upload/f/fc/ffcec6da4492c66099ef076537c2fcfa.jpg)
![](http://img.wesiedu.com/upload/5/f4/5f40ebf74c0455f7bc7362fcd03a97e9.jpg)
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.