# Π 1 1 x 4 2 1 y matb42h page 2 6 12 points a

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π . -1 1 x -4 -2 1 y

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MATB42H page 2 6. [12 points] (a) Evaluate integraldisplay γ ω , where ω = - y x 2 + y 2 dx + x x 2 + y 2 dy and γ ( t ) = (cos(3 t ) , sin(5 t )), 0 t 2 π . ( Hint: Carefully trace the path.) -1 1 x -1 1 y (b) Evaluate integraldisplay γ ( y +arctan( x 2 +1)) dx +( z + ye y 2 ) dy +ln( z 4 +1) dz , where γ consists of straight line segments from (2 , 0 , 0) to (0 , 2 , 0) to (0 , 0 , 4) and back to (2 , 0 , 0). 7. [12 points] Let S be the surface given by the parametrization Φ ( r, θ ) = ( r cos θ, 2 r cos θ, θ ) , 0 r 1 , 0 θ 2 π . (a) Find an equation for the tangent plane to S at Φ parenleftbigg 1 2 , π 3 parenrightbigg = parenleftbigg 1 4 , 1 2 , π 3 parenrightbigg . (b) Compute the surface area of S . 8. [6 points] Evaluate integraldisplay S z dS , where S is the piece of the cylinder x 2 + y 2 = 1 between the xy –plane and the plane z = 1 + x . -1 0 1 x -1 0 1 y 0 1 2 z 9. [32 points] (a) Find the flux of E ( x, y, z ) = ( x 2 + y 2 , 0 , x 3 y 2 + xz ) outward through the surface of the solid tetrahedron bounded by x + y + z = 1 and the coordinate planes. (b) The ellipse ( y - 4) 2 + 2 z 2 = 1 in the yz –plane is rotated about the z –axis to produce a surface S in R 3 . Determine the flux of F ( x ) = 1 bardbl x bardbl 3 x across S .
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