010.104-2005-2-mid - ; 2 o 5 > 2005 10 22{ 13r 15r 4...

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Unformatted text preview: ; 2 o 5 > 2005 10 22{ 13r 15r 4 Z 9 < V: s2: M K+ c a Z @\ %V IU lǣ k k( 100). \ 1. (10 points) Compute the length of the cardioid r = 1 + cos , given in the polar coordinates. 0 2 2. Justify your answers for the following questions. (a) (10 points) Is the vector field V(x, y, z) = (ex cos y + e-x sin z)i - ex sin yj + e-x cos zk a gradient field? (b) (10 points) Is the vector field W(x, y, z) = x(y 2 + 1)i + (yex - ez )j + x2 ez k the curl of some vector field? 3. Evaluate the following integrals: (a) (10 points) and y = x. (b) (10 points) 0 ey D 1 sin x dA, where D is the region bounded by x-axis, x = 1, x e x dxdy ln x xy 2 z 3 dV over the solid W bounded W 4. 5. (10 points) Compute the triple integral by z = xy, y = x, x = 1, and z = 0 in the octant x 0, y 0, z 0. (10 points) Find the volume of the region enclosed by the surfaces z = x2 + 3y 2 and z = 8 - x2 - y 2 . 6. 7. (10 points) Find the center of mass of the solid (of constant density) bounded above by the sphere x2 +y 2 +z 2 = a2 (a > 0) and below by the cone 3z 2 = x2 +y 2 . (10 points) The moment of inertia of a solid W about a line is defined by r2 dV, W where (x, y, z) is the density and r(x, y, z) is the distance to the line. Find the moment of inertia of a solid cylinder (of constant density ) with radius a and height h about its generators. (The generators of a cylinder are the lines that are tangent to the cylinder and parallel to its axis.) 8. (10 points) Compute the improper integral - - e-x 2 -2xy-3y 2 dxdy. ...
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010.104-2005-2-mid - ; 2 o 5 &amp;gt; 2005 10 22{ 13r 15r 4...

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