Homework Help: Rewrite Cartesian in Cylindrical form

1. Sep 28, 2016

nysnacc

1. The problem statement, all variables and given/known data

2. Relevant equations
Cartesian to Cylindrial

3. The attempt at a solution

What I was doing is that, I changed the limits of z, and the function.

2. Sep 28, 2016

Ssnow

you must be precise where $\rho$ varies. I give you an hint. Start from $-\sqrt{4-x^2-y^2}\leq z\leq \sqrt{4-x^2-y^2}$ and prove that $0\leq \rho \leq 2$...

3. Sep 28, 2016

nysnacc

So the whole volume is above and below the xy-plane... Can I divide the whole volume into 2 symmetric sub volume?

0≤ρ≤20≤ρ≤2 0\leq \rho \leq 2 as Φ is 0 where r = 0 :)

4. Sep 28, 2016

Staff: Mentor

Minor point, but your title (Rewrite Cartesian in Cylindrial form) and what you have in Relevant Equations, are incorrect. The problem asks you to transform the integral to spherical form, which is actually what you're doing. It might indicate that you don't have a clear understanding of the difference between cylindrical (not cylindrial, which I don't think is a word) coordinates and spherical coordinates.

5. Sep 28, 2016

nysnacc

θ

Okay, so am I setting the limits correctly in my work? thanks

6. Sep 28, 2016

Staff: Mentor

I don't think you are, but I haven't looked that closely at your work. Can you describe, in words, what the region of integration looks like?

7. Sep 28, 2016

nysnacc

In xy plane, the shape is a half circle in +x then z is bounded between -√ to +√ (so above and below xy plane)

Then r = x^2 + y^2 and θ goes from 0 to π, Φ goes from 0 to π as the volume is above and below the plane

8. Sep 28, 2016

Staff: Mentor

But what is the shape of the three-dimensional object that is the region of integration? That's what I'd like you to tell me. A very important aspect of being able to transform iterated integrals from one form to another is being able to correctly describe the region of integration. Once you understand this, evaluating the integrals is more-or-less mechanical.

9. Sep 28, 2016

nysnacc

A half sphere.

10. Sep 28, 2016

LCKurtz

You need to describe where the half sphere is located in the xyz coordinate system. You need to have the picture to get the limits correct. And, no, your limits in post #1 are not correct. For one thing, if the integration variable is $d\rho$, there cannot be any $\rho$ in the limits. And there is at least one other error.

11. Sep 28, 2016

nysnacc

Thanks.

By sketching, I come up with something like this:

θ: 0 to π
Φ: 0 to π
rho: 0 to 2 (the bound of the sphere)

And then I replace the function y^2 √ x^2+y^2+z^2 in terms of rho, θ and Φ

12. Sep 28, 2016

nysnacc

p is rho

∫ ∫ ∫ (p sinΦ sinθ)2 (p) p2sinΦ dp dΦ dθ ! ! !

13. Sep 28, 2016

LCKurtz

You haven't given us a description or picture of where the half sphere is located in the xyz coordinate system, and as a consequence your limits on $\theta$ are incorrect.

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