double integration of log(x^2+y^2)dxdy between 0toa and 0 to squareroot(a^2-y^2) a>0?

Question

double integration of log(x^2+y^2)dxdy between 0toa and 0 to squareroot(a^2-y^2) a>0

Answer 1

change to polar coordinates so that
x² + y² = r²
dxdy = rdrdq
r goes from 0 to a
q goes from 0 to Pi

=Int(Int(rlog(r²)drdq) r=0..a q = 0..Pi
=Pi Int(rlog(r²)dr r=0..a
u = r² du = rdr
= Pi Int(log(u)du) u = 0 .. a²
= Pi (u log(u) - u) u = 0..a²

you have to take the limit since log(0) is not defined,
you get

Pi(a²log(a²) - a²) - Pi(0 - 0)
=Pia²(log(a² - 1)

Answer 2

Convert to polar coordinates.