Order this complete solution of a classic: find the mass of a hemisphere of radius $R$ whose density grows with height, $\rho = kz$.

1. Recognize the symmetry and choose spherical coordinates
2. Express the density: $z = \rho\cos\varphi$, so density $= k\rho\cos\varphi$
3. Assemble the integrand with the Jacobian: $k\rho\cos\varphi \cdot \rho^{2}\sin\varphi$
4. Set hemisphere limits: $\rho \in [0,R]$, $\varphi \in [0, \pi/2]$, $\theta \in [0, 2\pi]$
5. Separate and evaluate the three one-variable integrals

This is the full multivariable pipeline in one problem: symmetry chooses coordinates, the Jacobian corrects volumes, limits encode the shape, and separability turns a 3D integral into three easy 1D ones (answer: $k\pi R^{4}/4$). Mastering the pipeline matters more than any single integral.