Fall 2014 Math 2E Review Problems Multiple Integrals cid 90 cid 90 cid 90 Problem 1 Evaluate x y 2 and the cylinder y2 z2 1 in the rst octant E z dV where E is bounded by the planes y 0 z 0 cid 90 cid 90 cid 90 z3 cid 113 Problem 2 Evaluate that lies above the xy plane and has center the origin and radius 1 x2 y2 z2 dV where H is the solid hemisphere H Problem 3 Find the volume of the solid tetrahedron with vertices 0 0 0 0 0 1 0 2 0 and 2 2 0 the half cone z cid 112 x2 y2 Problem 4 Find the volume of the solid above the paraboloid z x2 y2 and below Problem 5 Use the transformation x u2 y v2 z w2 to nd the volume of the region bounded by the surface z 1 and the coordinates planes x y 13 24 14 2 3 6 1 90 Problem 6 Use an appropriate change of variables to evaluate where R is the square with vertices 0 2 1 1 2 2 and 1 3 cid 90 cid 90 x y x y R dA ln 2 1 Problem 7 Use spherical coordinates to evaluate cid 90 2 cid 90 2 0 4 y2 cid 90 4 x2 y2 4 x2 y2 y2 cid 113 x2 y2 z2 dzdxdy Problem 8 Rewrite the integral as an iterated integral in order dxdydz 64 9 cid 90 1 cid 90 1 cid 90 1 y 1 x2 0 cid 90 1 cid 90 1 z 0 0 cid 90 y y f x y z dzdydx f x y z dxdydz Vector Calculus cid 90 Problem 9 Evaluate 1 1 C x ds where C is the arc of the parabola y x2 from 0 0 to Problem 10 Evaluate 0 t yz cos x ds where C x t y 3 cos t z 3 sin t and Problem 11 Evaluate 1 0 1 to 3 4 2 C xy dx y2 dy yz dz where C is the line segment from Problem 12 Evaluate given by r t t2 cid 126 i t3 cid 126 j t cid 126 k 0 t 1 C F dr where F x y z ez cid 126 i xz cid 126 j x y cid 126 k and C is cid 90 C cid 90 cid 90 5 1 1 12 5 6 10 110 3 2 11 12 4 e Problem 13 Show that F is conservative and use this fact to evaluate F x y 4x3y2 2xy3 cid 126 i 2x4y 3x2y2 4y3 cid 126 j and C r t t sin t cid 126 i 2t cos t cid 126 j 0 t 1 F dr where Problem 14 Show that F is conservative and use this fact to evaluate F dr where F x y ey cid 126 i xey ez cid 126 j yez cid 126 k and C is the line segment from 0 2 0 to 4 0 3 cid 90 C cid 90 C Problem 15 Use Green s theorem to evaluate cid 112 cid 90 C 1 x3 dx 2xy dy where C is the triangle with vertices 0 0 1 0 and 1 3 Problem 16 Find curlF and divF if F x y z e x sin y cid 126 i e y sin z cid 126 j e z sin x cid 126 k Problem 17 Find an equation of the tangent plane at the point 4 2 1 to the parametric surface S given by where 0 u 3 and 3 v 3 r u v v2 cid 126 i uv cid 126 j u2 cid 126 k x 4y 4z 0 0 2 3 3 Problem 18 Find the area of the part of the surface z x2 2y that lies above the triangle with vertices 0 0 1 0 1 2 1 6 27 5 cid 90 cid 90 5 32 3 cid 90 cid 90 S Problem 19 Evaluate the surface integral the plane z 4 x y that lies inside the cylinder x2 y2 4 S x2z y2z dS where S is the part of Problem 20 Evaluate the surface integral F dS where F x y z x2 cid 126 i xy cid 126 j z cid 126 k and S is the part of the paraboloid z x2 y2 below the plane z 1 with upward orientation Problem 21 Use Stoke s theorem to evaluate cid 82 cid 82 2 S curlF dS where F x y z x2yz cid 126 i yz2 cid 126 j z3exy cid 126 k where S is the part of the sphere x2 y2 z2 5 that lies above the plane z 1 and S is oriented upward Problem 22 Use Stoke s theorem to evaluate cid 82 4 C F dr where F x y z xy cid 126 i yz cid 126 j zx cid 126 k and C is the triangle with vertices 1 0 0 0 1 0 and 0 0 1 oriented counter clockwise as viewed from above Problem 23 Use the Divergence theorem to calculate the surface integral cid 82 cid 82 S F dS where F x y z x3 cid 126 i y3 cid 126 j z3 cid 126 k and S is the surface of the solid bounded by the cylinder x2 y2 1 and the planes z 0 and z 2 1 2 4 Problem 24 Compute the outward ux of F x y z x cid 126 i y cid 126 j z cid 126 k x2 y2 z2 3 2 through the ellipsoid 4x2 9y2 6z2 36 Problem 25 If a is a constant vector r x cid 126 i y cid 126 j z cid 126 k and S is an oriented smooth surface with a simple closed smooth positively oriented boundary curve C show that 2a dS a r dr cid 90 cid 90 S 11 4 cid 90 C 5
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