How does the equation x^2 + y^2 = 1 form a cylinder?

Question

amistre64 · Answer

you strech it along the z axis

turingtest · Answer

On a plane this is a circle.
You have to specify that we are in 3-space, and that $z$ can vary freely if you want to describe a cylinder
so perhaps a more accurate representation of this equation in $\mathbb R^3$ is$$x^2+y^2=1,z\in \mathbb R$$

turingtest · Answer

which is a slightly more fancy way of saying what amistre said :)

anonymous · Answer

so would you say then that z = 1- x^2 - y^2?

turingtest · Answer

no, that equation would be a parabaloid I think
we just want a cylinder
here is a drawing of$$x^2+y^2=1$$in the xy-plane
(I put the xy-plane on its side like it's lying down)
|dw:1333642959046:dw|

turingtest · Answer

now what if I include the z-axis?|dw:1333643226125:dw|we keep the same restrictions on x and y (that they have to form a circle) but now z can take on any value along that resreiction
so it "extends" the shape along z

turingtest · Answer

|dw:1333643329979:dw|those circles are all supposed to be the same shape :/
and the cylinder goes off to infinity in both directions

anonymous · Answer

because I have an equation of a cylinder z = 16 - x^2 which doesn't make sense to me... I know that cylinders can have either the z axis as its center or the y or x axis as its center, but shouldn't it then be z^2 = 16 - x^2?

turingtest · Answer

z = 16 - x^2 is a parabola
if y is allowed to vary freely I'm not sure what kind of shape you get, but it ain't a cylinder
you are right that if it were  z^2 = 16 - x^2 and y varied freely we would have a cylinder along the y-axis

turingtest · Answer

...with radius 4

anonymous · Answer

Oky I'm going to give you the whole problem:
Find the volume of the solid in the first octant bounded by the cylinder z = 16 - x^2 and the plane y = 5.

turingtest · Answer

yeah that makes no sense to me

turingtest · Answer

if the typo is just supposed to be z^2=16-x^2 the problem is easy
otherwise I don't know what they are talking about

amistre64 · Answer

if y varies freely its called a parabolic cylindar

anonymous · Answer

Well the answer is: $$\int\limits_{0}^{4}\int\limits_{0}^{5}(16 - x ^{2})dydx$$ I understand why they use the 16 - x^2 and the integral of y, but I don't know where the integral of x comes from...

turingtest · Answer

ah....
amistre to the rescue :D

amistre64 · Answer

|dw:1333644473814:dw|

amistre64 · Answer

z=16-x^2 = 0 when x = 4 or -4

amistre64 · Answer

|dw:1333644598298:dw|