S={(x ,y ,z) ЄR³ : x² + y² ≤ (2-z)² , 0 ≤ z ≤ 1}

with variable density(x, y, z)= -ln(z)/√x² + y²

I have no clue how to do this...please help...(Crying)

Printable View

- July 17th 2012, 07:10 AMangel050382Calculate the mass M of the truncated cone
S={(x ,y ,z) ЄR³ : x² + y² ≤ (2-z)² , 0 ≤ z ≤ 1}

with variable density(x, y, z)= -ln(z)/√x² + y²

I have no clue how to do this...please help...(Crying) - July 17th 2012, 07:42 AMHallsofIvyRe: Calculate the mass M of the truncated cone
Surely you know that the mass of an object is its density integrated over its volume.

Your volume is bounded by the cone , below by z= 0 and above by z= 1. The only thing peculiar about this is that the vertex of the cone is at z= 2. Since z<1 2- z is below 2 and we can use . Projecting onto the xy-plane, we get circle with center at (0,0) and radii 1 to 2.

In x, y, z coordinates, we would have to integrate over the 'washer' between the circle with radius 1 and the circle with radius 2. That can be done but would require four separate integrals (from x= -2 to x= -1, from x= -1 to 1 with y above the smaller circle, from x= -1 to 1 with y below the smaller circle, from x= 1 to 2). It is much easier to use cylindrical coordinates. In that case, r goes from 1 to 2, [itex]\theta[/itex] goes from 0 to 2\pi, and, for each r and [itex]\theta[/itex], z goes from the plane, z= 1 up to the cone, [tex]z= 2-\sqrt{x^2+ y^2}= 2- \sqrt{r^2}= 2- r.

In cylindrical coordinates the density function is .

That is your integral is . - July 17th 2012, 07:48 AMReckonerRe: Calculate the mass M of the truncated cone
- July 17th 2012, 07:55 AMReckonerRe: Calculate the mass M of the truncated cone
- July 17th 2012, 09:18 AMHallsofIvyRe: Calculate the mass M of the truncated cone
Good point. I think I lost track in the middle and started thinking I was only doing the outside! Thanks for the correction.

- July 17th 2012, 09:30 AMReckonerRe: Calculate the mass M of the truncated cone
- July 17th 2012, 09:34 AMangel050382Re: Calculate the mass M of the truncated cone
Thank you guys so much!!!! I really appreciate your help. I have another question!

How should I go on now? Cuz if I start to integrate I have problem at ln(0) what is not defined! How should I handle this now!

I know I have a lot of stupid questions, but I really try to pass that math class...:-( - July 17th 2012, 11:16 AMtom@ballooncalculusRe: Calculate the mass M of the truncated cone
Just in case a picture helps work the integral from the inside out...

http://www.ballooncalculus.org/draw/intMulti/five.png

... where (key in spoiler) ...

__Spoiler__:

__________________________________________________ __________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote! - July 17th 2012, 11:49 AMReckonerRe: Calculate the mass M of the truncated cone