Volume integral in cylindrical coordinates

[whatisreality]

Homework Statement

OK, I thought once I knew what the question was asking I'd be able to do it. I was wrong!

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.

Homework Equations

The Attempt at a Solution

I've sketched both equations, so I have a vague idea of what's going on. The z=... equation is an 'elliptic paraboloid', the other eqn a cylinder. I could find the volume by finding the volume of the cylinder and subtracting the volume above the paraboloid, although the limits for that might be complicated. That's what I'd do in 2D, never done a 3D question.

I think I have to find the z coordinate where they intersect, but I don't know how. And I'm supposed to convert to cylindrical, which is probably the only bit I can at least try, using x=rcos(θ), y=rsin(θ) and z=z:

r²cos²(θ) + r²sin²(θ) = 4R²
And
(r²cos²(θ) + 3r²sin²(θ) )/R = z

I think I could replace r with 2R? Then the equations become
cos²(θ) + sin²(θ) = 1
And
4Rcos²(θ) + 12Rsin²(θ)=z
Not sure where to go now!

 

[Ray Vickson]

Homework Statement

OK, I thought once I knew what the question was asking I'd be able to do it. I was wrong!

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.

Homework Equations

The Attempt at a Solution

I've sketched both equations, so I have a vague idea of what's going on. The z=... equation is an 'elliptic paraboloid', the other eqn a cylinder. I could find the volume by finding the volume of the cylinder and subtracting the volume above the paraboloid, although the limits for that might be complicated. That's what I'd do in 2D, never done a 3D question.

I think I have to find the z coordinate where they intersect, but I don't know how. And I'm supposed to convert to cylindrical, which is probably the only bit I can at least try, using x=rcos(θ), y=rsin(θ) and z=z:

r²cos²(θ) + r²sin²(θ) = 4R²
And
(r²cos²(θ) + 3r²sin²(θ) )/R = z

I think I could replace r with 2R? Then the equations become
cos²(θ) + sin²(θ) = 1
And
4Rcos²(θ) + 12Rsin²(θ)=z
Not sure where to go now!

Your integration region $R$ is 3-dimentisonal:
$$R = \{ (x,y,z): x^2 + y^2 \leq 4 R^2, 0 \leq z \leq (x^2 + 3 y^2)/R$$
In cylindrical coordinates $x = r \cos \theta, y = r \sin \theta$ these become
$$\begin{array}{l} r^2 \leq 4 R^2 \Longrightarrow 0 \leq r \leq 2 R \\ 0 \leq z \leq r^2 (\cos^2 \theta + \sin^2 \theta + 2 \sin^2 \theta)/R = r^2 (1 + 2 \sin^2 \theta)/R \\ 0 \leq \theta \leq 2 \pi \end{array}$$

 

[whatisreality]

Your integration region $R$ is 3-dimentisonal:
$$R = \{ (x,y,z): x^2 + y^2 \leq 4 R^2, 0 \leq z \leq (x^2 + 3 y^2)/R$$
In cylindrical coordinates $x = r \cos \theta), y = r \sin \theta)$ these become
$$\begin{array}{l} r^2 \leq 4 R^2 \Longrightarrow 0 \leq r \leq 2 R \\ 0 \leq z \leq r^2 (\cos^2 \theta + \sin^2 \theta + 2 \sin^2 \theta)/R = r^2 (1 + 2 \sin^2 \theta)/R \\ 0 \leq \theta \leq 2 \pi \end{array}$$

OK, I can follow that. And then the function I actually perform the volume integral on would be the z=... function, with those limits?

 

[Ray Vickson]

OK, I can follow that. And then the function I actually perform the volume integral on would be the z=... function, with those limits?

You tell me.

 

[whatisreality]

You tell me.

Those are definitely the limits of the integration. And the function to be integrated obviously has to be in cylindrical coordinates too. So I think the function should be the z= function in cylindrical coordinates.

 

[whatisreality]

Because if I integrated the other function, that would just give me the volume of the cylinder. But I'm still not really confident with my answer!

 

[Ray Vickson]

Because if I integrated the other function, that would just give me the volume of the cylinder. But I'm still not really confident with my answer!

I am not sure what your issue is here. What was the answer you got? What detailed formulas did you use to get it? Did you do it two different ways and get two different answer? What, exactly, is the problem?

 

[whatisreality]

I am not sure what your issue is here. What was the answer you got? What detailed formulas did you use to get it? Did you do it two different ways and get two different answer? What, exactly, is the problem?

Oh, no, haven't computed the integral yet. My issue was, initially, I didn't know what to do, which I now do. The answer I'm not confident with is the answer that I should be integrating the z= function, even though I'm nearly 100% sure it's right. But I'm nearly never confident in my answers, so that's fine! :)

I do know how to actually do the integral now that I have the limits though. Triple integral with respect to r dr dθ dz. I really appreciate you taking the time to reply by the way, thank you for your help!

 

[whatisreality]

I am not sure what your issue is here. What was the answer you got? What detailed formulas did you use to get it? Did you do it two different ways and get two different answer? What, exactly, is the problem?

Well, now I have done the integral, and I do actually still have a problem...

So since V=∫∫∫ r dr dθ dz, you integrate r dr dθ dz with the limits found?
Although actually, you have to rearrange the z= function to get an expression for r. Which would be

r = $\sqrt{ \frac{zR}{1+2 \sin^2(\theta)}}$

Which I would not enjoy integrating. Or do you just make a function f(r, θ, z) by turning z = $\frac{r^2(1+2\sin^2(\theta))}{R}$ into

f(r, θ, z) =$\frac{r^2(1+2\sin^2(\theta))}{R} - z$? I don't actually know what to integrate.

 

[whatisreality]

Er, not just ∫∫∫ r dr dθ dz, I think that might be just for cylinders actually.