Sunday, 7 June 2015

integration - Double integrals in polar coordinates



Determine the domain of D={(x,y)R2|x[12,12],y[|x|,1x2]} in polar coordinates and draw it.




Also how would you integrate D11+x2+y2dA which is i guess
12121x2|x|11+x2+y2dydx



I guess in the integral you can use the polar coordinates
D11+r2cos2(ϕ)+r2sin2(ϕ)rdrdϕ
Dr1+r2drdϕ
34π14π10r1+r2drdϕ=34π14π(12ln(1+12)12ln(1+02))dϕ
34π14πln22dϕ=(34πln2214πln22)=π4ln2



Did I get it right?



Answer



Draw a picture. A simple plot reveals that the domain D is simply the sector of the circle r=1 between two values of θ. A little thought provides those values of θ (i.e., what purpose does the absolute value serve?).



The integrand you show is also wrong, as 1+r22.



The answer I get is (π/4)log2.


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