Hey so I am having trouble with some parametrization questions. Whenever there's a cylinder with a plane that isn't strictly horizontal or vertical I get messed up:

Need help with 1(ii) and just the parametrization part of 6, computing the integrals should be no problem:




For 1ii) I just tried

x = rcost
y = rsint
z = x + 3 = rsint + 3

With 0 <= r <= 1 and 0 <= t <= 2pi

I was just wondering if this gets the the whole elliptic disk or just the ellipse.

For 6) I can't find a parametrization for all the surfaces. For the circle at y = 0 I just have (x,y,z) = (rcost, 0, rsint) with r in [0,1] and t in [0,2pi]. And for the other two surfaces I get hung up. I think the parametrization for both the sliced cylinder and the elliptic disk at the front of the solid is (x,y,z) = (rcost, 2 - rcost, rsint) with r in [0,1] and t in [0,2pi]. Would they be the same? If my 1(ii) is correct then I guess they should be different but visually I can't see it.
Also the question does not specify which normals I should be using so should I choose the ones that are all the consistent? i.e. with all normals pointing outward or all pointing inward?

Any help is appreciated

For 1 (ii), your answer is correct (well it's z = r.cos t + 3), and yes you'll get the whole disk, since r is free to take any value between 0 and 1 (take r = 1 and then you'll have only the boundary : the ellipse, only 1 parameter remains).

6) Correct for the disk at y = 0.
(r.cos t ; 2 - r.cos t ; r.sin t) with r in [0;1] and t in [0;2.pi] stands for a parametrization of the elliptic disk on the plane x + y = 2.
( cos t ; y ; sin t ) with t in [0;2.pi] and y in [0;2-cost] would be a parametrization for the sliced cylinder.

For the cylinder if I made it

(cos t ; s(2 - cos t) ; sin t) with t in [0,2pi] and s in [0,1] would that be the same as what you said?

Yes, exactly the same.

Perfect, thanks feanur!!