PP2 - 1 A at plate is dened by the area between the curves...

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1. A flat plate is defined by the area between the curves y = x 2 and y = x . The density of the plate is f ( x, y ) = 1. (a) Draw the plate (be sure to label everything properly). (b) Find the mass of the plate. (c) Find the center of mass for the plate. (d) Set up the integral for the polar moment of inertia. 2. Use the following function to answer the questions. f ( x, y ) = x 3 - 3 x 2 - 2 y 3 + 3 y 2 (a) Find the direction of maximum increase at the point P = (1 , 0 . 5). (b) What is the rate of change in this direction? (c) Find a vector that is tangent to the level curve at point P. (d) Find and categorize the local extrema of the function. (e) Find the equation for the plane tangent to the surface at point P. 3. Fully justify your answers. (a) Evaluate the integral using any appropriate method integraldisplay 2 0 integraldisplay 4 - x 2 0 e x 2 + y 2 dydx (b) Set up but DO NOT SOLVE an integral that describes the mass of a spherical shell that lies above cone 3 z 2 = x 2 + y 2 . The sphere has an outer radius of 3 meters, a thickness of 1 meter, and
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