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 ??