Determine the domain of D={(x,y)∈R2|x∈[−1√2,1√2],y∈[|x|,√1−x2]} in polar coordinates and draw it.
Also how would you integrate ∫∫D11+x2+y2dA which is i guess
∫1√2−1√2∫√1−x2|x|11+x2+y2dydx
I guess in the integral you can use the polar coordinates
∫∫D11+r2cos2(ϕ)+r2sin2(ϕ)rdrdϕ
∫∫Dr1+r2drdϕ
∫34π14π∫10r1+r2drdϕ=∫34π14π(12ln(1+12)−12ln(1+02))dϕ
∫34π14πln22dϕ=(34πln22−14πln22)=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+r2≠2.
The answer I get is (π/4)log2.
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