I know how to deal with the axis part, but I struggle solving the integral with the curve $$x^3 + y^3 = x^2 + y^2$$ I tried to parametrize the curve using polar coordinates. $$x = r(t)\cos(t)$$ $$y = r(t)\sin(t)$$
where $$r(t) = \frac{1}{\cos^3(t)+\sin^3(t)}$$ Using $$\iint_S 1 \,d(x,y) = \int_{\partial S} x\,dy = $$ $$ \int_0^{\pi/2} \frac{\cos(t)}{\cos^3(t)+\sin^3(t)} \cdot\frac{\cos(t)(\cos^3(t)+\sin^3(t))+\sin(t)(3\cos^2(t)\sin(t)-3\sin^2(t)\cos(t))}{(\cos^3(t)+\sin^3(t))^2} \, dt $$ $${}+\int_{\pi/2}^\pi 0 \, dt + \int_\pi^{2\pi} 0 \, dt$$
But I can't solve the first integral. I tried to find another parametrization but wasn't succesful. I'm sure there is a simplier way to solve this than computing the integral. Can you give me some advice about the easier parametrization or hint how to solve the integral?