Problems: Green’s Theorem and Area 1. Find M and N such that M dx + N dy equals the polar moment of inertia of a uniform C density region in the plane with boundary C. Answer: Let R be the region enclosed by C and ρ be the density of R. The polar moment of inertia is calculated by integrating the product mass times distance to the origin: ll ll ll I= dI = r2 dm = (x2 + y 2 ) · ρ dA. R R R Green’s theorem now tells us that we’re looking for functions M and N such that Nx −My = ρx2 + ρy 2 . The simplest choice is Nx = ρy 2 , My = −ρx2 . This leads to N = ρxy 2 , M = −ρx2 y. Use Green’s theorem to check this answer: 2 ll 2 ρy 2 − (−ρx2 ) dA −ρx y dx + ρxy dy = C l lR = R r2 · ρ dA = I.

Problems: Green’s Theorem and Area

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