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Unformatted text preview: 5 . Triple Integrals 5A. Triple integrals in rectangular and cylindrical coordinates ~d~ lx 5A-1 Evaluate: a) b) l2 ' 2xy2z dz dx dy 5A-2. Follow the three steps in the notes to supply limits for the triple integrals over the following regions of 3-space. a) The rectangular prism having as its two bases the triangle in the yz-plane 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 x-coordinate 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 y-axis, and base the quarter circle in the xz-plane 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. 5A-3 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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