How to Set Up a Surface Integral for Finding Average Value on a Unit Sphere?

[jacksonb62]

Homework Statement

Find the average value of the function f(x,y,z)=xyz on the unit sphere in the first octant

Homework Equations

I know that I need the surface integral of xyz over the sphere and then need to divide by the surface area of the region, but I'm having a hard time setting up the integration. We have always done surface integrals with double integrals, but for this I feel like spherical coordinates need to be used which I only know how to set up in a triple integral. Help please!

The Attempt at a Solution

 

[LCKurtz]

Homework Statement

Find the average value of the function f(x,y,z)=xyz on the unit sphere in the first octant

Homework Equations

I know that I need the surface integral of xyz over the sphere and then need to divide by the surface area of the region, but I'm having a hard time setting up the integration. We have always done surface integrals with double integrals, but for this I feel like spherical coordinates need to be used which I only know how to set up in a triple integral. Help please!

The Attempt at a Solution

Spherical coordinates with $\rho$ constant will only have two variables, giving a double integral. What is the formula for the element of surface area in spherical coordinates? $dS = ??$.