Sheet4 - Math 21C DHC Konba Discussion Sheet 4 1 Consider...

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Unformatted text preview: Math 21C DHC Konba Discussion Sheet 4 1.) Consider the tetrahedron R with vertices (2,0,3), (0,0,3), (0,4,0), and (0,4,3). a.) Sketch R in three dimensional space. b.) Describe R using rectangular coordinates by first projecting R onto i.) the (Icy-plane. ii.) the yz—plane. iii.) the xz-plane. c.) SET UP but DO NOT EVALUATE a double integral which represents the volume of R. (1.) SET UP but DO NOT EVALUATE a triple integral which represents the volume of R. «7r/2 z y 2.) Evaluate / f / cos($ + y + z) dzr dy dz . 0 0 0 .. 3.) Evaluate the following integrals by first converting to polar (cylindrical) coordinates. 1 :v 1 3/2 9— 11:2 2 a)/ / ————dyd:r b./O /3w /\/$2+y2dzdyd$ 1N5 m¢x2+y2 4.) Sketch the solid in three dimensional space whose volume 1s given by the following integral. 77/2 'CSC9 1:21— 2—- \/:1:—2——y2 a.) / / (5r—r HcosO—r sin0) drdfi b)./11/\/1T{2/ ldzdyda: .n/4 0 222+y2 5.) Consider the rectangular box in three dimensional space bounded by the planes 1: = 0, a: = 4, y— — 0, y— — 3, z— — 0, and z -— 5. Assume that the density at the point P = (5r, y, z) is numerically equal to 7 plus the distance from P to the point (3, 4, 5). SET UP but DO NOT EVALUATE a triple integral which represents the moment of inertia of this solid about a.) the origin. b.) the z—axis. 6.) The center of mass of a solid R of mass M is located at (0,0,0). Its moment of inertia about the m—axis is I. a.) Find the moment of inertia for R about a line parallel to the m—axis and k units from the :12—axis. b.) About which such line parallel to the az-axis is the moment of inertia the least ?...
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