Parameterize union of two shapes in 3space

Question

I am supposed to parameterize the union of the two shapes $x^2 + y^2 = 1, z = y$.

I do not even know how to get the union of the two shapes. When I graph the two shapes the intersection does not correspond to $z=\left(1-x^2\right)^{1/2}$, which is what I get when I set both equations equal to $y$, then each other. I am then supposed to find the line integral along the curve of intersection. How is there a curve of intersection? Is the intersection not a 2-D circular surface?

Answer 1

I assume you mean intersection, not union, since you're finding a line integral along the curve of intersection. In this case, you'd just parametrise $x^2+y^2=1$, and give $z$ the same value as $y$. A possible way to do this would be $(\cos(\theta), \sin(\theta), \sin(\theta))$ on $0 \leq \theta < 2\pi$.

Answer 2

Your $$x^2+y^2=1$$is a cylinder shell not a solid one.

When you cut it with a slant plane the intersection is an ellipse.

You may parametrize the intersection with $$x= \cos(t), y=\sin(t), z=\sin(t)$$