Friday, December 23, 2016

integration - Converting double integrals to polar form: what would the limits be here?


So I have the following integral:


x2dxdy(1+x2+y2)5


I know that converting the integral using polar coordinates gives:


r2cos2θ(1+r)5rdrdθ


I'm assuming r is going from 0 to infinity.


But what about θ?


Answer



Note that in order to cover R2, r extends from 0 to and θ spans an entire period of sin(θ) and cos(θ). (For example, the first quadrant alone is covered by θ[0,π/2], r[0,).)



Hence,


x2(1+x2+y2)5dxdy=2π00r2cos2(θ)(1+r)5rdrdθ=(2π0cos2(θ)dθ)=π(0r3(1+r)5dr)=1/4=π4


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