Finding the equation of a 3 dimensional surface

[ElijahRockers]

Homework Statement

Find the equation of the surface generated by rotating the line x = 9y about the x-axis.

The Attempt at a Solution

I am not sure how to go about this one... I know that x=9y is a line in the xy plane that crosses through the origin, so rotating it about the x-axis will create a double cone, but I don't know how to find its equation.

And hints would be appreciated

 

[jedishrfu]

We can't solve the problem but we can give hints

what is the equation for the surface area of a cone? what values do you need to compute the surface area from this formula?

draw a picture of x = 9y rotated about the x-axis and identify those values and it should become obvious.

 

[ElijahRockers]

Oops. Wrong forum, this should be in Calculus I guess. I am in Cal 3, I put it here because there is really no calculus involved in this question, it is just three dimensional geometry.

Anyway, I already know the answer, it's

$x^2 = 81y^2 + 81z^2$

I'm just not sure how to arrive at that. I tried to study this on my own last semester, but I still have a difficult time with these 3d objects.

Anyhow, the equation for a surface area of a cone would be geometrically the same as the surface area of a circle, just with a little piece of the pie missing. I'm not trying to find the area of the object, I'm trying to find the equation that represents the object.

I drew a little 3D picture, and from looking at it, I know that if I were to take cross-sections with yz planes they would look like bigger circles the further away from x=0 I got.

Cross sections with xy planes would look like hyperbolas except at z=0, where it would just be the line x=9y.

Cross sections with xz planes would look almost, if not identical to the xy planes.

I also know the general form for any surface in R3 I think:

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

That's if my memory is correct.

 

[A. Bahat]

Upon inspection, that "general equation" for a surface in ℝ³ can't be right: it's not even a surface! It's a conic section, a very specific kind of curve, in ℝ². If you added another variable, say z, to that equation, you'd get the general form of a 2-dimensional quadric (the higher-dimensional analogue of a conic). This still leaves out many possible surfaces, e.g. z=x³+y³.

As for the original problem, here's my hint: use the Pythagorean theorem.

 

[ElijahRockers]

Oh, sure enough. Obviously I've got some serious issues regarding 3D objects, haha...
Back to the text, I suppose.

 

[HallsofIvy]

It should be clear from the start that rotating a line around an axis gives a cone.

I would set up parametric equations with r= y and $\theta$ the angle rotated around the x axis. The equation x= 9y becomes $x= 9r$ while y and z being in the plane of rotation, $y= rcos(\theta)$ and $z= rsin(\theta)$. We can go back to Cartesian coordinates by noting that
$$y^2+ z^2= r^2= x^2/81$$

 

[ElijahRockers]

Interesting, I will take a look at it from that angle, thank you so much!