ryrt - C is the intersection of the plane x + 2 y + z = 5...

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MATH 241H - EXAM 4 - December 6, 2004 No calculators. Show your work. [15] 1. Evaluate R C y 2 dx +2 x dy + yz dz where C is the line segment starting at (1 , 0 , - 1) and ending at (3 , 4 , - 2). [20] 2. Evaluate R C (sin x - y 3 ) dx + ( x 3 + e y ) dy where C is the circle x 2 + y 2 = 4 oriented clockwise. [25] 3. Let Σ be the surface parametrized by r ( u, v ) = (1 + u ) i + (3 - v ) j - uv k with u 2 + v 2 9. (a) Find an equation of the tangent plane to Σ at the point r (2 , 1). (b) Find the surface area of Σ. [20] 4. Evaluate R C F · d r where F ( x, y, z ) = ( x 2 + z ) i + (3 x - y 2 ) j + (5 y - z 2 ) k and
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Unformatted text preview: C is the intersection of the plane x + 2 y + z = 5 and the cylinder x 2 + y 2 = 9 oriented counterclockwise viewed from above. [20] 5. Evaluate R R F n dS where F ( x, y, z ) = (2 x + y ) i +(4 y + x 2 ) j +( x 2 y 2 + z ) k and is the entire surface of the solid region bounded above by the paraboloid z = 1-x 2-y 2 and bounded below by the xy-plane, with outward pointing normal vectors....
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