Triple integration w/spherical coordinates

[MAins]

1.

"Find the mass of part of the solid sphere x^2 + y^2 + z^ 2 ≤ 25 in the 1st octant x ≥ 0, y ≥ 0, z ≥ 0 where mass density is f (x, y, z ) = (x^2 + y^2 + z^2 )^3/2 ."

3.

These problems are really stumping me! I need somebody to work it out/explain it to me! What will the limits of integration be for the following question? What do i integrate? I know I need to transform it to spherical coordinates... but beyond that I'm lost.

I know it's a triple integral:
m = ∫∫∫(x^2 + y^2 + z^2)^3/2 dzdydx
transforming to spherical co-ordinates:
0 ≤ rho ≤ 5
0 ≤ theta ≤ ? (how do I figure this out?)
0 ≤ phi ≤ ? (ditto)

dzdydx = rho^2 sinphi drho dphi dtheta
What does f (x, y,z) transform to and how do I figure it out?

 

[Dick]

x^2+y^2+z^2=rho^2. What does that make f(x,y,z)? phi is the polar angle and theta is the equatorial angle. What range of these keeps you in the first octant? Refer to a picture of your preferred system of spherical coordinates.

 

[tiny-tim]

… keep it simple …

Hi MAins!

You're making this too complicated …

It's one-eight of the mass of the whole sphere.

Divide it into spherical shells, of radius r, from r = 0 to 5.

Then the mass of each shell is … ?

 

[HallsofIvy]

But just in case, you would like to learn how to do these problems!

Since $\rho= \sqrt{x^2+ y^2+ z^2}$, $(x^2+ y^2+ z^2)^{3/2}= \rho^3$. Thats the function you want to integrate.

Yes, you are correct that the differential of volume in spherical coordinates is $\rho sin^2(\phi)d\rho d\theta d\phi$

Now, to determine what the limits of integration should be, think about what the variables in spherical coordinates mean. $\rho$ measures the distance from the origin, (0,0,0) to a point. Since your sphere is centered at (0,0,0) and has radius 5, $\rho$ must go from 0 to 5. $\theta$ measures the angle around the "equator" (think of it as "longitude"). For the full sphere, it goes from 0 to $2\pi$. Here, you have just 1/4 of a full circle: so $\theta$ goes from 0 to ?? $\phi$ measures the angle down from the z-axis to the point (think of it as "co-latitude". For the full sphere, it goes from 0 to $\pi$. You only want to go from the z-axis to the xy-plane, 1/2 way: so $\phi$ goes from 0 to ??