Finding Mass of Hemisphere z = √(4-x²-y²) with ρ = z

Calculus 3 · Axiom Academy

EXAMPLE Mass of the Hemisphere Find the mass of a surface by integrating its density with the scalar surface integral . A thin shell has the shape of the upper hemisphere (radius 2 , so ). Its area density is — the shell is heaviest at the top and vanishes at the rim. Find the total mass . The shell is the top half of a sphere of radius 2 ; we compute its mass by projecting the surface integral straight down onto the base disk D . Nicely done. You found the mass of a curved shell by turning a surface integral into an ordinary double integral over its shadow. Scalar surface integral: mass is , the density summed over the surface's own area element dS . Project to the base: for a graph z=f(x,y) , . Forgetting this Jacobian factor is the classic mistake — . The hemisphere shortcut: for the factor collapses to , so with the z 's cancel and . The same "flatten the surface onto its shadow" move powers flux integrals, centers of mass, and moment-of-inertia computations across all of vector calculus.

This is the written version of the interactive lesson above. See the full Calculus 3 course.