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For the volume case, the integrand is constant (implicitly 1), so integrating with respect to one variable (like $z$) simply multiplies the area of the base by the height, reducing it to a double integral. For the mass problem with variable density $\rho(x,y,z)=xyz$, the density changes with $z$, so the integration must be performed over all three dimensions to accumulate the varying mass contributions.

Conditions: The goal is volume rather than mass with variable density.; The integrand over the volume is constant for the volume case.; The density varies with position for the mass case.

For the volume case, the integrand is constant (implicitly 1), so integrating with respect to one variable (like $z$) simply multiplies the area of the base by the height, reducing it to a double integral. For the mass problem with variable density $\rho(x,y,z)=xyz$, the density changes with $z$, so the integration must be performed over all three dimensions to accumulate the varying mass contributions.

Conditions: The goal is volume rather than mass with variable density.; The integrand over the volume is constant for the volume case.; The density varies with position for the mass case.