How should I evaluate this line Integral : $$ \int_{C} (y^2 + z^2)dx + (z^2 + x^2)dy + (x^2 + y^2)dz$$ where $C$ is the part for which $z \ge 0$ of the intersection of the surfaces $x^2 + y^2+ z^2 = 4x$, $x^2+ y^2 = 2x$ and the curve begins at origin and runs at first in the positive octant.
My main Problem Is I am not able to visualize how the curve is formed due to which neither I can define the parametric coordinates nor I can setup the integral properly.
Can anyone tell me how should I solve this problem ?
Hint. The curve is the intersection of the sphere $(x-2)^2+y^2+z^2=2^2$ and the cylinder $(x-1)^2+y^2=1^2$ contained in the half-space $z\geq 0$.
Let $x(t)=1+\cos(t)$, and $y(t)=-\sin(t)$ with $t\in [-\pi,\pi]$. Then $(x-1)^2+y^2=1^2$ is satisfied.
Now are you able to find $z(t)$ and write the intersection curve $t\to (x(t),y(t),z(t))$ in a parametric form?