Question

Evaluate ∫∫ln(x^2+y^2)/sqrt(x^2+y^2) dA where 1≤x^2 +y^2≤e

Answer 1

This looks like a mess, but in polar coordinates we have
r^2 = x^2+y^2
dA = r dr dθ

so it gets really simple

∫∫ln(r^2)/r r dr dθ
where 1<=r^2<=e, 0<=θ<=2π
∫∫2 ln(r) dr dθ
∫[0,2π] 2r(ln(r)-1) dθ [1,√e]
∫[0,2π] (2-√e) dθ
2π(2-√e)