Final Unknown

Final Unknown - MATH 52 FINAL EXAM 1(a Sketch the region R...

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MATH 52 FINAL EXAM 1. (a) Sketch the region R of integration in the following double integral. integraldisplay 1 0 integraldisplay 1 x xe y 5 dy dx (b) Express the region R as an x -simple region. (c) Evaluate the integral by changing the order of integration. 2. (a) Let T be a solid cone whose base is the disk x 2 + y 2 1 in the xy -plane and whose vertex is the point (0 , 0 , 2). Set up, but do not evaluate, a triple integral in cylindrical coordinates which computes the moment of inertia of T around the x -axis. (b) Let T be the solid ball of radius 1 centered at (2 , 0 , 0). Set up, but do not evaluate, a triple integral in spherical coordinates which computes the moment of inertia of T around the z -axis. Hint: First consider the change of variables u = x - 2, v = y , w = z . 3. Consider the curve C parametrized by r ( t ) = (2 e t , 1 3 e 3 t + e - t ) for 0 t 1. (a) Find the arclength of C . (b) Find the x -coordinate of the centroid of C . (c) Let S be the surface obtained by rotating C around the line x = - 1. Find the area of S .

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