## calculus – Using the parameterization for a curve in the xy-plane to calculate the line integral

My question is:

Find a parameterization for the curve in the $$xy$$-plane given by the parabola with equation $$x = y^2.$$

Using this, calculate the line integral $$int_C mathbf{F}(x)cdot ds$$, where $$C$$ is the curve given by the parabola $$x = y^2$$ from $$(0, 0)$$ to $$(9, 3)$$, and $$mathbf F$$ is the vector-valued function (or vector field) $$mathbf{F}(x,y) = (x^2,y^2) = x^2mathbf{i}+y^2mathbf{j}$$.

I understand what the question is asking for but I have no idea where to begin and how to proceed. Would extremely appreciate any help!

## calculus – Normal line calculation from a given surface to the xy-plane

Question: Suppose you head toward the $$xy$$-plane from the surface $$x^2+y^2-z^2=-1$$ at the point $$(1, 1, sqrt{3})$$ by following the normal line to the surface at that point. What are the $$x$$ and $$y$$ coordinates at which you will hit the $$xy$$-plane?\$

My attempted Solution: I know that the normal line has the parametric equation

$$r(t) = <1+2t, 1+2t, sqrt{3}+2sqrt{3}t>$$
but I’m unsure of how to find the $$xy$$ coordinates at which this line hits the $$xy$$ plane. I tried setting $$z=0$$ and having $$sqrt{3}+2sqrt{3}t = 0$$. In this case, $$t=-1/2$$. Solving for $$x$$ and $$y$$, they are somehow also both zero.

Could someone please show me how this is done? Thanks!