Integral, mass and polar coordinates

Complexity · 2014-05-29 10:48:24

I have been given a calculus question in which I can't find the same result as my teacher.
"An object T in space is bounded by

and , and the density of T is ."

I've tried to rewrite the boundaries to polar-coordinate regions, such that

, and , such that:
and

I tried to calculate the mass:

However my teacher has the answer 4pi, so where did I go wrong?

Last edited by Complexity (2014-05-29 10:49:20)

gAr · 2014-05-29 19:19:33

What are the bounds of x and y?

Complexity · 2014-05-29 21:45:35

I've given you all the information that I've been given: The 2 z-boundaries and the density.
However I would think we're working with the boundary:

Complexity · 2014-05-29 23:23:17

Oh well I get the result now by saying:

However the reason for the density to stay 'z' and then make the integral run from 0 to r is to me a bit confusing. Can anyone explain why this is the right way?

Last edited by Complexity (2014-05-30 02:02:37)

gAr · 2014-05-29 23:36:00

I assumed various bounds, but got neither of the answers.

And in 3 dimensions, you need to consider cylindrical or spherical coordinates for transformation. I did not understand what you did there.

Complexity · 2014-05-30 00:14:30

Well as I wrote, I posted all the information I was given (I translated the question), so I can't give you any more information.
If it is true that I need to consider either cylindrical or spherical coordinates in 3 dimensions, then I don't have to make this exercise, because our exam won't contain any of this, however this is one of the exercises mentioned we should be able to make. - I am kind of confused myself, so what I did was trying to calculate the result through polar coordinates.

Last edited by Complexity (2014-05-30 00:15:00)

gAr · 2014-05-30 00:44:45

Okay, then I think the source of the question itself is missing some information.
I think it's doable without any transformation.

Complexity · 2014-05-30 00:53:48

I hope the question is not missing any information since it is from an earlier exam. This might be too much asking of you, but could you try to calculate it without transformation? - I'm personally lost, and my biggest challenge with these exercises are those tricky boundaries.

gAr · 2014-05-30 01:27:33

Getting the bounds in 3D is tricky for me as well, can't imagine the shape precisely!

I am thinking transformation may not be necessary because if we write the integral:

the square root will be eliminated on integrating w.r.t z.

Complexity · 2014-05-30 02:16:48

I see why you were having problems when writing the problem like that. To imagine the shape i tend to use the polar coordinates on a problem like this, because:

, hence it indicates that the xy-plane is a disc with changing radius, however a radius can not be smaller than 0 and the shape is also bounded by the the value of 2 for the radius. We haven't bound the disc from entering any of the quadrants (i hope that is the english term) of the xy-plane, so we can let it run from 0 to 2pi.
By this information I imagine it is a cone. However i still don't get the part where:
.
It is bothering me why I can't let z be an r, since I know the density = z = r. I don't know why I can't put the boundaries as I have done. So the only part I actually know about this integral is why we multiply by r (it is because it is polar coordinates).
- The entire thing just seem to give the correct result, but why? I am clueless.

gAr · 2014-05-30 03:02:35

Hi,

I remembered it wrong, I looked up the cylindrical coordinates again.
You missed the limits slightly.
I think I got the answer now:

Complexity · 2014-05-30 03:16:08

Thank you so much for your help!
Seems like it was cylindrical coordinates afterall, meaning either we should solve it in another way (doubtable) or our teacher simply forgot to point out, that this is one of the exercises we should not make.
Anyway now I can make some progress, so once again thank you

Last edited by Complexity (2014-05-30 03:18:33)

gAr · 2014-05-30 03:21:25

Glad to help you, you are welcome!

Math Is Fun Forum

#1 2014-05-29 10:48:24

Integral, mass and polar coordinates

#2 2014-05-29 19:19:33

Re: Integral, mass and polar coordinates

#3 2014-05-29 21:45:35

Re: Integral, mass and polar coordinates

#4 2014-05-29 23:23:17

Re: Integral, mass and polar coordinates

#5 2014-05-29 23:36:00

Re: Integral, mass and polar coordinates

#6 2014-05-30 00:14:30

Re: Integral, mass and polar coordinates

#7 2014-05-30 00:44:45

Re: Integral, mass and polar coordinates

#8 2014-05-30 00:53:48

Re: Integral, mass and polar coordinates

#9 2014-05-30 01:27:33

Re: Integral, mass and polar coordinates

#10 2014-05-30 02:16:48

Re: Integral, mass and polar coordinates

#11 2014-05-30 03:02:35

Re: Integral, mass and polar coordinates

#12 2014-05-30 03:16:08

Re: Integral, mass and polar coordinates

#13 2014-05-30 03:21:25

Re: Integral, mass and polar coordinates

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