Mass of a sphere, checking my result

**arbolis** · July 20th 2009, 04:46 PM

Let $B$ be a ball with radius $a$ and in which the density of any point is equal to the distance to a fixed diameter. Find the mass of the ball.

My attempt : $m=\iiint _{B} \rho (r,\phi, \theta) drd\phi d \theta = \int_0^{2\pi} \int_0^{\pi} \int _0^a r^3 \sin (\phi) drd\phi d\theta=a\pi$ .
I had a hard time finding that $\rho (x,y,z)=x^2+y^2$ if I chose the fixed diameter as the $z$ -axis. Then I simply converted into spherical coordinates chosing " $r$ " instead of " $\rho$ " because I already got confused in another exercise because of the notation.
Is my result correct?

**Random Variable** · July 20th 2009, 05:30 PM

Do they mean the shortest distance?

If that's what they mean, the shortest distance from a point to the z-axis is $\sqrt{x^{2}+y^{2}} = r \sin \phi$ .

**Calculus26** · July 20th 2009, 05:36 PM

See attachment--I think you are also confusing the cylindrical coordinate r.

**arbolis** · July 20th 2009, 05:46 PM

Originally Posted by Random Variable

Do they mean the shortest distance?

If that's what they mean, the shortest distance from a point to the z-axis is $\sqrt{x^{2}+y^{2}} = r \sin \phi$ .

I guess yes.
Ah yes, I knew it was $\sqrt{x^2+y^2}$ but I don't know why I wrote $x^2+y^2$ , even in my draft!

Originally Posted by Calculus26

See attachment--I think you are also confusing the cylindrical coordinate r.

Oops, right. Ahmm... poor me.
Calculus 26, $\delta (x,y,z)=\sqrt{x^2+y^2}$ as Random Variable pointed out.

**Calculus26** · July 20th 2009, 05:52 PM

thanks that makes the integrand p^3 sin^2(phi) etc...

Thread: Mass of a sphere, checking my result

LinkBack

Thread Tools

Display

Mass of a sphere, checking my result

Similar Math Help Forum Discussions

Solving for Mass, Given Mass as Constant + Infinite # of Discrete Unknown variables)

please help with using double integrals to solve for mass and center of mass

[SOLVED] Find a residue (checking my result and clearing doubts)

[SOLVED] Integral, checking a result

[SOLVED] Checking my result (infinite series)

Search Tags