It looks like you want to compute the double integral
[tex]\displaystyle \iint_D (x+y) \,\mathrm dx\,\mathrm dy[/tex]
over the region D with the unit circle x ² + y ² = 1 as its boundary.
Convert to polar coordinates, in which D is given by the set
D = {(r, θ) : 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π}
and
x = r cos(θ)
y = r sin(θ)
dx dy = r dr dθ
Then the integral is
[tex]\displaystyle \iint_D (x+y)\,\mathrm dx\,\mathrm dy = \iint_D r^2(\cos(\theta)+\sin(\theta))\,\mathrm dr\,\mathrm d\theta \\\\ = \int_0^{2\pi} \int_0^1 r^2(\cos(\theta)+\sin(\theta))\,\mathrm dr\,\mathrm d\theta \\\\ = \underbrace{\left( \int_0^{2\pi}(\cos(\theta)+\sin(\theta))\,\mathrm d\theta \right)}_{\int = 0} \left( \int_0^1 r^2\,\mathrm dr \right) = \boxed{0}[/tex]
