Is x^2 + y^2 = 4R^2 a Cylinder or a Circle in 3D Space?

[whatisreality]

I don't think this goes in the homework section because I don't actually want help answering the question, I want to know what it means!

Consider the volume V inside the cylinder x² +y² = 4R² and between z = (x² + 3y²)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant. Write down a triple integral for the volume V using cylindrical coordinates. Include the limits of integration (three upper and three lower). Evaluate the integral to determine the volume V in terms of R.

My main problem is when it asks about the cylinder x² +y² = 4R². I'm nearly 100% sure that equation is not actually for a cylinder but for a circle! And I'm not entirely clear on whether I'm integrating two shapes, as in two volume integrals, or it's describing just one big shape.

In the latter case, I still don't know where cylinders come into it.

 

[Incand]

The equation is for a circle but you also have that $z$ varies between the $xy$-plane and $(x^2+3y^2)/R$. If you stack a lot of circles on top of each other you get a cylinder. So the first equation only describe one part of the cylinder while the third coordinate, $z$ is free too change value.
So the equation is a circle if you were in a plane, If you were in 3d-space you have an infinite cylinder if you didn't have any restrictions on $z$.

 

[Daeho Ro]

I draw the shape of that cylinder with $R=1$.

 

[Mark44]

Consider the volume V inside the cylinder x² +y² = 4R² and between z = (x² + 3y²)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant.

The equation is for a circle but you also have that $z$ varies between the $xy$-plane and $(x^2+3y^2)/R$. If you stack a lot of circles on top of each other you get a cylinder. So the first equation only describe one part of the cylinder while the third coordinate, $z$ is free too change value.
So the equation is a circle if you were in a plane, If you were in 3d-space you have an infinite cylinder if you didn't have any restrictions on $z$.

In three dimensions (which is implied by the statement that x, y, and z are coordinates), the equation ($x^2 + y^2 = 4R^2$) is a right circular cylinder. Since z does not appear in the equation, it is arbitrary.

 

[whatisreality]

In three dimensions (which is implied by the statement that x, y, and z are coordinates), the equation ($x^2 + y^2 = 4R^2$) is a right circular cylinder. Since z does not appear in the equation, it is arbitrary.

Arbitrary as opposed to zero?

 

[Mark44]

Arbitrary as opposed to zero?

"Arbitrary" means "any value."

 

[whatisreality]

"Arbitrary" means "any value."

I know. So arbitrary means it can take the value zero and others, as opposed to just zero, which is what I thought the equation meant. Shouldn't the z appear somewhere in the equation though? I feel like this is quite a basic concept I've misunderstood or missed! Oops!

 

[Mark44]

I know. So arbitrary means it can take the value zero and others, as opposed to just zero, which is what I thought the equation meant. Shouldn't the z appear somewhere in the equation though? I feel like this is quite a basic concept I've misunderstood or missed! Oops!

No, z doesn't have to appear in the equation. In the plane, the equation x = 2 is a vertical line. Here, y is not mentioned, and it is arbitrary, so every point in the plane with coordinates (2, y) is a point on this line. The situation is similar for your cylinder equation.

 

[whatisreality]

Oh, I get it! Ok, that is a really important thing to know. Also know what the question is asking now! Thanks for your help :)