1 Let R denote the shaded region pictured below right Compute 12y dA 4 points R 1 1 y y 2 x x y2 R 2 0 x R 12y dA 2 Completely setup but do not evaluate a triple integral giving the volume of the pyramid shown at right This pyramid has a square base and the triangular faces lie in the planes given by x z 1 x z 1 y z 1 and y z 1 5 points z 0 0 1 1 1 0 1 1 0 y x Volume 1 1 0 Z Z 1 Z r2 3 For each of the integrals a 0 0 0 f r z r dz dr d and b Z 0 p Z 1Z 0 0 z f r z r dr dz d label the solid corresponding to the region of integration below 2 points each z z z y y y x x x z z z y y y x x x D E 2 4 Let F x y x 1 cos y 2x e y Let R denote the solid semi disk shown below right Let C denote the y boundary of the region R Z a Use Green s Theorem to evaluate C F d r where C C has the orientation shown 3 points R x 1 0 1 0 Z C F dr b Let D denote the part of the curveZC above consisting only of the semicircle not the line segment with the orientation shown Compute D F d r 3 points Z D F dr 5 Let R be the region shown at right y v 1 2 2 2 T S 1 1 y 2 R xy 2 y 1 2 1 xy 1 u x a Find a transformation T R2 R2 taking S 1 2 1 2 to R 3 points T u v D E y 2 dA via an integral over S 4 points 1 Emergency backup transformation if you can t do a pretend you got the answer T u v u 5 v u and do part b anyway b Use your transformation T u v from part a to evaluate R R y 2 dA 6 Let S be the surface parameterized by r u v v cos u v v sin u for 0 u 2 and 0 v 1 a Mark the picture of S below 2 points z z z y y x z x x y x y y dS 6 points b Evaluate the surface integral S S y dS p 7 Let E be the solid above the cone z x 2 y 2 and below the plane z 2 Use spherical coordinates to completely setup but not evaluate a triple integral which computes the volume of E 4 points 8 For each surface S in parts a and b give a parameterization r D S Be sure to explicitly specify the domain D and call your parameters u and v a The portion of the sphere x 2 y 2 z 2 1 where x 0 3 points D n u D r u v v and D n u D r u v E b The triangle in R3 with vertices 2 0 0 0 1 0 0 0 1 which lies in the plane x y z 1 2 points 2 v and o o E c Parameterize the cylinder C x 2 y 2 1 by r u v cos u sin u v for 0 u 2 and v unrestricted Let M be the part of C above the x y plane and below the plane x z 2 Find a region D in R2 so that r D M 1 point D 0 u 2 and v d Let M be the surface in part c Is the surface integral x dS M Circle your answer 1 point negative zero positive
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