1 Let R be the region of integration pictured below right Evaluate R 6y dA 4 points y 2 1 1 1 R 0 0 1 0 R x 6y dA 2 Set up but DO NOT EVALUATE a triple integral that computes the volume of the tetrahedron shown at right 5 points 1 0 1 z z x x y 1 x 1 0 0 0 0 0 0 1 0 y 3 Set up but DO NOT EVALUATE a triple integral that computes the volume of the region that lies inside the sphere x 2 y 2 z 2 2 and above the cone z x 2 y 2 5 points 4 Consider the triple integral gration below 3 points 1 2 1 y 1 x 2 y 2 0 y 0 f x y z dz dx dy Mark the corresponding region of intez z z y x x y z y x z z x x x y y y 5 Find a transformation T R2 R2 which takes the unit circle to the ellipse given by x 3 2 shown 3 points y2 1 as 4 y v 2 2 T 1 1 2 1 1 2 u 1 1 1 1 2 2 2 T u v 3 4 x 6 Let R be the region in the x y plane depicted below right Let T u v 2u v u v a Find a rectangle S in the uv plane whose image under T that is the collection of points T u v for all choices of u v in S is exactly R 3 points y 2 1 x R ANSWER S u v b Set up but DO NOT EVALUATE the integral R 4 1 2 2 u v cos x dA as an integral in the u v coordinates If you can t do part a leave the limits of integration blank 5 points R 2 2 cos x dA 2 2 x cos y x dA 0 where R is the region shown below x y 7 Let F x y x x cos y Then R Compute F d r where C is the pictured curve that goes from 3 0 to 3 0 via 0 2 3 points C 0 2 3 0 3 0 8 Consider the region D in the plane bounded by the curve C as shown at right For each part circle the best answer 1 point each 2 a For F x y x 1 y the integral F d r is C y C negative zero positive b The integral negative C y dx 2 dy is zero D positive x c The integral negative D y x dA is zero positive 9 For each surface S below give a parameterization r D S Be sure to explicitly specify the domain D and call your parameters u and v a The rectangle in R3 with vertices 1 0 0 0 1 0 1 0 2 0 1 2 3 points z 1 0 2 D u and r u v v b The portion of cone y 0 1 2 y 0 1 0 1 0 0 x x 2 z 2 for 0 y 1 which is shown at right 4 points z y x D r u v u and v 10 Consider the surface S parameterized by r u v v 2 u v for 0 u 1 and 0 v 1 a Mark the correct picture of S below 2 points z z z z x y y y y b Evaluate the integral x x x S z dA 6 points S z dA 11 Consider the solid described as follows using cylindrical coordinates E is the region inside the paraboloid z 1 r 2 and where 0 and z 0 Choose one double integral and one triple integral below that compute the volume of E 1 point each 1 1 y 2 1 x 2 y 2 1 1 z 1 dz dx dy 2 1 z y 2 dy dz 1 y 2 0 1 1 z 1 z x 2 0 0 1 z 0 1 dy dx dz 1 1 y 2 1 x 2 y 2 0 1 y 2 0 1 dz dx dy 0 0 0 0 1 1 z 2 2 1 y 2 z 2 dy dz 1 1 z 2 1 z 2 y 2 d y dz 0 0
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