Find $\int_C{ \vec{F} \cdot \vec{dr}},$ where $F(x, y, z) = \langle 2 x^2 y , 2 x^3 /3, 2xy\rangle$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z = y^2 – x^2$ and the cylinder $x^2 + y^2 = 1$, oriented counterclockwise as viewed from above.
I tried to parametrize the region but have no idea what to do after that.
Best Answer
In the comments we reduced this to integrating $\langle 2x, -2y,0\rangle$ over the part of paraboloid $z=y^2−x^2$ that lies within the cylinder $x^2+y^2=1$. Note that the Stokes theorem was already used, the rest is just a run-of-the-mill computation of a surface integral. So if you are uncertain about how this is done, that would be the topic to revisit. Some points to keep in mind