We can find the moment of inertia of a lamina in the plane with density (x, y) by

Chapter 5, Problem 30

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We can find the moment of inertia of a lamina in the plane with density (x, y) by considering the lamina to be a flat plate sitting in the x y-plane in R3. Then, for example, the distance of a point (x, y) in the lamina to the x-axis is given by |y|, the distance to the y-axis is given by |x|, and the distance to the z-axis (or the origin) is given by x 2 + y2. (See Figure 5.123.) Using these ideas, find the specified moments of inertia of the laminas given in Exercises 2931. x z y (x, y) |x| | y| x2 + y2 Figure 5.123 A lamina situated in the x y-plane in R3.The moment of inertia Iz about the z-axis of the lamina shaped as the rectangle [0, 2] [0, 1], and whose density varies as (x, y) = 1 + y.