$$\int_D (xy)\mathrm{d}x-x^2\mathrm{d}y+z\mathrm{d}z$$ where $D:\ x^2+y^2-4z^2=1 ; z=1$
I can't seem to figure out how to make the figure nor what are the parametrics, I thought they would be: $x(t)=\sqrt5\cos t ; y(t)=\sqrt5\sin t ; z(t)=1$ where $t \in[0,2\pi]$ I think.
I should be able to solve this problem after I learn how to figure these 2 things out, any help is much appreciated!
Ok, I am guessing you mean that you are cofined to move in the circular cross section of the surface and plane $z=1$. We can get the cross section by plugging $z=1$ into the surface's equation:
$$ x^2 + y^2 = 5$$
Returning to line integral:
$$ \int_D (xy)dx - x^2 dy + z dz$$
Note that since we don't move in the z direction, the last integral is zero. This leads to:
$$ \int_D (xy)dx - x^2 dy $$
After this you can parameterize as with $ x =\sqrt{5} \cos(t) $ and $y= \sqrt{5}\sin(t)$