1. The problem statement, all variables and given/known data The problem asks for a single triple integral (the integrand may be a sum but there must be a single definition for the bounds of the integral) representing the volume (in the first octant) of the shell defined by a sphere of radius 2 centered around the origin and a sphere of radius 1 centered around (0,0,1) in spherical coordinates. dV must be in the order dρdφdθ. 2. Relevant equations For the large sphere, 4 = x^2 + y^2 + z^2, and for the smaller one 1 = x^2 + y^2 + (z+1)^2. 3. The attempt at a solution I know the bounds for φ, which are 0 to π/2. The bounds for θ are the also 0 to π/2. The integrating factor is ρ^2sinφ, for dV dρdφdθ. What I don't understand is how to define ρ. What is particularly confusing is that when 0 ≤ φ ≤ ≈ 26° there is no single radius that extends from the origin to the edge of the shell. You can't draw a straight line from origin to outer surface without going through the smaller sphere. Surprisingly, I was unable to find a similar problem posted here or anywhere else. If you know of one, I would greatly appreciate that link. Thank you very much for any input you have to offer.