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Unformatted text preview: 5 . Triple Integrals 5A. Triple integrals in rectangular and cylindrical coordinates ~d~ lx 5A1 Evaluate: a) b) l2 ' 2xy2z dz dx dy 5A2. Follow the three steps in the notes to supply limits for the triple integrals over the following regions of 3space. a) The rectangular prism having as its two bases the triangle in the yzplane cut out by the two axes and the line y + z = 1, and the corresponding triangle in the plane x = 1 obtained by adding 1 to the xcoordinate of each point in the first triangle. Supply limits for three different orders of integration: (iii) /// dy dx dz b)* The tetrahedron having its four vertices at the origin, and the points on the three axes where respectively x = 1, y = 2, and z = 2. Use the order /// dz dy dx. c) The quarter of a solid circular cylinder of radius 1 and height 2 lying in the first octant, with its central axis the interval 0 5 y 5 2 on the yaxis, and base the quarter circle in the xzplane with center at the origin, radius 1,and lying in the first quadrant. Integrate with respect to y first; use suitable cylindrical coordinates. d) The region bounded below by the cone z2 = x2 + y2, and above by the sphere of radius 4 and center at the origin. Use cylindrical coordinates. 5A3 Find the center of mass of the tetrahedron D in the first octant formed by the coordinate planes and the plane x + y + z = 1. Assume 6 = 1....
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This note was uploaded on 09/18/2008 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.
 Spring '08
 Auroux
 Integrals, Limits

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