Finding COM of Triangle with Density ρ(x,y) = x+y

Calculus 3 · Axiom Academy

EXAMPLE Center of Mass of a Triangle with Density Locating the balance point of a lamina whose density grows with x and y A thin triangular plate occupies the region with vertices (0,0) , (2,0) , and (2,2) — the triangle bounded by y=0 , x=2 , and y=x . Its density is . Find the center of mass , where and . The plate is denser toward the top-right corner, so its balance point sits right of the geometric center. Nice work. You found the center of mass of a triangular plate whose density varies across the region. Here's the playbook. Describe the region once: the triangle bounded by y=0 , x=2 , and y=x is , — the same bounds serve M , M_y , and M_x . Mass before moments: compute first, since both coordinates divide by it. Moments weight the density: (distance from the y -axis) and (distance from the x -axis). Because grows toward the top-right corner, the balance point is pulled that way — right of where a uniform plate of the same shape would balance.

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