ps5 - University of Hong Kong Fall 2011 Math 1813...

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Unformatted text preview: University of Hong Kong Fall 2011 Math 1813 Mathematical Methods for Actuarial Science Instructor: Siye Wu Problem Set 5 (due at 17:00, Friday, 25 November) I. Evaluate the following integrals. 1 1 2 (a) 0 1 e−y dy dx. x√ 1−y 2 (b) 0 0 (x + y ) dxdy , where D is the region x2 + y 2 (c) D 1 0 z x + y. z 1 (d) 0 (x2 − xy + y 2) dx dy dz . (3 + z 2 ) dxdydz , where D is the solid hemispherical dome given by x2 + y 2 + z 2 (e) z 1 dxdy . 1 + x2 + y 2 0. 4 and D xy √ dxdydz , where D is the solid region in the first octant bounded by the cone z 2 = z D x2 + y 2 and the planes x = 0, y = 0, z = 1. (f) II. Find the area or volume, and the centre of mass, of each of the following regions. (a) the planar region bounded by the curves y = x2 , y = 2x2 , x = y 2 and x = 2y 2. (b) the solid tetrahedron whose vertices are (0, 0, 0)T , (a, 0, 0)T , (0, b, 0)T and (0, 0, c)T , where a, b, c > 0 are constants. (c) the solid region bounded by the sphere x2 + y 2 + z 2 = 2az (where a > 0 is a constant) and the cone x2 + y 2 = z 2 . [Note: The centre of mass of a planar region D of area A, assuming uniform density, is the point (¯, y )T ∈ R2 , where x = A−1 D x dxdy and y = A−1 D y dxdy . The centre of mass of a solid x¯ ¯ ¯ region in space is similarly defined.] ...
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