Use Stokes' Theorem to evaluate the line integral $$ F = -y^3 dx + x^3 dy - z^3 dz$$ where C is the intersection of the cylinder$$ x^2 + y^2 = 1$$ and the plane $$x+y +z=1$$

I solved it but I have 1 doubt in the answer given in the book.

SOLUTION:

$$
\operatorname{curl} F = 3\big(x^2+y^2\big) \,\hat{k} $$

Then taking projection on $x-y$ plane, eventually we get this answer (as per me)

$$ \int\_0^{2\pi}\int\_0^1\sqrt3r^2 r\,dr\,d\theta. $$

The solution in book says that because $x^2+y^2=1$ (because they lie on the circle), therefore, the answer would be $$ \int\_0^{2\pi}\int\_0^1\sqrt3 r\,dr\,d\theta. $$ 

But I think because we are talking about the area of circle, therefore $x,y$ don't lie on the circle.