If L is the circle which you get from the intersection between the sphere $$ x^2+y^2+z^2=1, y=x\sqrt(3) $$ and $$ I= \int_L (y-z)dx+(z-x)dy+(x-y)dz $$ so |I| equals to?
but i dont understand how the intersection is a circle if i compare between them im getting $$ 4x^2+z^2=1 $$ but this thing is an ellipse what am i doing wrong?
Your confusion arises because the equations of conic sections in 2D and 3D are different. It is not enough to write $4x^2+z^2=1$ to specify an ellipse in 3D - in fact, that is the equation of a cylinder with an elliptical cross-section, and the y-axis as cylinder axis. However, if you add $y=0$, it becomes an ellipse.
Now imagine cutting this elliptic cylinder with a plane. If you use planes parallel to the xz plane by picking a constant y, you'll get ellipses. But if you cut the elliptic cylinder with a plane of just the right inclination to the xz plane you can actually specify a circle in space.