{0 to 1} ∫ {y to 1} ∫ x dxdy

We can think of Cartesian coordinate x as a projection of polar coordinate r to x axis:
x=r cosθ
so we'll substitute this in our Cartesian integral.
Besides that, while in Cartesian coordinates elementary area dA was
dA=dxdy
in polar coordinates it is
dA=r drdθ
so instead of x dxdy under integral sign, we write
(r cosθ) r dr dθ = r² cosθ dr dθ

Now about limits of integration:
The domain of integration is right triangle bounded by x-axis and lines x=1 and y=x.
In this domain, in polar coordinates, angle θ changes from 0 to π/4, while r changes from r=0 (=origin) to r=sec θ (=equation of line x=1 in polar coordinates).

Finally, our polar integral is:
{0 to π/4} ∫ {0 to secθ} ∫ r² cosθ dr dθ