Evaluating Triple Integrals on a Sphere in the First Octant

[amiras]

Homework Statement

Evaluate triple integral

z^2 dxdydz

throughout
i) the part of the sphere x^2 + y^2 + z^2 = a^2 (first octant)
ii)the complete interior of the sphere x^2 + y^2 + z^2 = a^2 (first octant)

Homework Equations

It is probably good idea to work in spherical coords.

z = r*cosφ
x = r*sinφ cosθ
y = r*sinφ sinθ

dxdydz = r^2 sinφ drdφdθ

The Attempt at a Solution

I'l start at part ii) because its the part I can do.
Here the boundaries are:
0 =< r < a
0 =< φ < pi/2
0 =< θ < pi/2

the integration now becomes:

(Int[r=0, a] r^4 dr )( Int[φ=0, pi/2] sinφcos^2 φ)( Int [θ=0, pi/2]) = r^5/30 * pi

i) But for part i), I am confused. The integral should be evaluated only on the surface of the sphere. The radius a is constant in length, so how should r be defined?
a < r < a, makes no sense.

Need advice.

 

[SteamKing]

Are you sure the problem is quoted correctly?

 

[LCKurtz]

Part i obviously means the interior of the sphere in the first octant. Otherwise it wouldn't be a triple integral.

 

[amiras]

You are right, I understand what nonsense I was thinking about. The part i) is not quoted correctly. Thanks!