1. Let R be the region of integration pictured below right. Evaluate�R6y dA. (4 points)xy(0,0)(1,0)(2,1)(−1,1)R�R6y dA =2. Set up, but DO NOT EVALUATE, a triple integral that computes the volume of the tetrahedron shown atright. (5 points)xyz(0,0,0)(1,0,0)(0,1,0)(1,0,1)x + y =1z = x3. Set up, but DO NOT EVALUATE, a triple integral that computes the volume of the region that lies inside thesphere x2+y2+z2=2 and above the cone z =�x2+y2. (5 points)4. Consider the triple integral�1/20�1−yy�1−x2−y20f (x, y, z) dz dx dy. Mark the corresponding region of inte-gration below. (3 points)xyzxyzxyzxyzxyzxyz5. Find a transformation T : R2→ R2which takes the unit circle to the ellipse given by (x −3)2+y24= 1 asshown. (3 points)� 2� 112� 2� 112uvT� 11234� 2� 112xyT (u, v) =�,�6. Let R be the region in the xy-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 allchoices of (u, v) in S) is exactly R. (3 points)xy(2,−2)(4,−1)(2,1)RANSWER: S =�(u, v)����≤u ≤ , ≤ v ≤�.(b) Set up, but DO NOT EVALUATE, the integral�Rcos(x) dA as an integral in the (u, v)-coordinates. Ifyou can’t do part (a), leave the limits of integration blank. (5 points)�Rcos(x) dA =7. Let F(x, y) =�x2, x2cos(y)�. Then�R�∂∂x�x2cos(y)�−∂∂y�x2��dA = 0 where R is the region shown below.Compute�CF ·dr, where C is the pictured curve that goes from (−3,0) to (3,0) via (0,2). (3 points)(−3,0) (3,0)(0,2)8. Consider the region D in the plane bounded by the curve C as shown at right. For each part, circle the bestanswer. (1 point each)(a) For F(x, y) =�x +1, y2�, the integral�CF ·dr isnegative zero positive(b) The integral�C�−y dx +2dy�isnegative zero positive(c) The integral�D(y −x)dA isnegative zero positiveDCxy9. For each surface S below, give a parameterization r : D →S. Be sure to explicitly specify the domain D andcall your parameters u and v.(a) The rectangle in R3with vertices (1,0,0),(0,1,0),(1,0,2),(0,1,2). (3 points)xyz(1,0,0)(0,1,0)(1,0,2)(0,1,2)D =�≤u ≤ and ≤ v ≤�r�u, v�=�,,�(b) The portion of cone y =�x2+z2for 0 ≤ y ≤1 which is shown at right. (4 points)xyzD =�≤u ≤ and ≤ v ≤�r�u, v�=�,,�10. Consider the surface S parameterized by r(u, v) =(v2, u, v)for0≤u ≤1 and 0 ≤ v ≤1.(a) Mark the correct picture of S below. (2 points)xyzxyzxyzxyz(b) Evaluate the integral�Sz dA. (6 points)�Sz dA =11. Consider the solid described as follows using cylindrical coordinates: E is the region inside the paraboloidz = 1 −r2and where 0 ≤ θ ≤ π and z ≥ 0. Choose one double integral and one triple integral below thatcompute the volume of E. (1 point each)�10��1−y2−�1−y2��1−x2−y201 dz dx dy�10��1−z02�1 −z − y2dy dz�10��1−z−�1−z��1−z−x201 dy dx dz�10��1−z202�1 −y2−z2dy dz�10��1−y2−�1−y2�1−�x2+y201 dz dx dy�10�1−z02�(1 −z)2−y2d y
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