e304k

e304k - Exam III Key MAC 2312 Nov 21, 2004 S Hudson 1) The...

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Exam III Key Nov 21, 2004 MAC 2312 S Hudson 1) The plane x + y + z = 12 intersects the paraboloid z = x 2 + y 2 in an ellipse. Find the highest and lowest points on this ellipse. For full credit, set this up and solve it as a Lagrange multiplier problem with two constraints. 2) Compute R R R x dx dy , where R is the plane region bounded by y = x 2 and y = 8 - x 2 . 3) Solve by double integration in polar coordinates: Find the volume bounded by the paraboloids z = x 2 + y 2 and z = 4 - 3 x 2 - 3 y 2 . 4) Find the area cut from the saddle-shaped surface z = xy by the cylinder x 2 + y 2 = 1. 5) Use cylindrical coordinates to ﬁnd the centroid of the solid bounded by the plane z = 0 and the paraboloid z = 9 - x 2 - y 2 . 6) Answer True or False. You do not have to explain. The Jacobian of T can be negative. If E is a solid of revolution around the z axis, then its moment of inertia is a point (0,0,a) on the z axis. If the density of an object is constant, then its mass equals its volume.

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This note was uploaded on 12/27/2011 for the course MAC 2313 taught by Professor Grantcharov during the Fall '06 term at FIU.

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e304k - Exam III Key MAC 2312 Nov 21, 2004 S Hudson 1) The...

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